Reliable Multicast Transport E. Stauffer
Internet Draft Broadcom
B. Shen
Broadcom
S. Chakraborty
Broadcom
D. Tujkovic
Broadcom
J. Huang
Broadcom
S. Shet
Broadcom
K. Rath
Broadcom
Intended status: Standards Track March 29, 2014
Expires: September 2014
Supercharged Codes
draft-stauffer-rmt-bb-fec-supercharged-04.txt
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Abstract
This document describes a fully-specified FEC scheme for the
Supercharged forward error correction code. Supercharged codes are
designed for use on the erasure channel. Coding for the erasure
channel commonly arises for data transmission over the internet, where
lower layers either successfully deliver packets or fail to deliver
them. Coding is required to insure that data is not lost, even if
packets are lost at the lower layers. Error free reception is
important for multimedia applications, such as streaming, where it may
not be possible to correct an error in time by any other means. Coding
insures that lost packets can be recovered.
Table of Contents
1. Introduction...................................................3
2. Supercharged Code..............................................3
2.1.1. Definitions..........................................3
2.2. Overview..................................................4
2.3. Matrix Representation.....................................5
2.4. Systematic Encoding.......................................6
2.5. Erasure Channel...........................................6
2.6. Decoding..................................................7
2.7. Matrix P Construction.....................................7
2.7.1. Function Prototypes..................................7
2.7.2. Parallel Filter Code T Construction..................8
2.7.3. Repetition Code R Construction......................10
2.7.4. Block Code B_1 Construction.........................11
2.7.5. Block Code B_2 and B_3 Construction.................11
2.7.6. SC_Parameters.......................................13
2.7.7. K Table.............................................13
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2.7.8. Random Number Generator.............................18
2.7.9. Random Permutation..................................22
2.7.10. RS Generator.......................................23
2.7.11. RS Code............................................24
2.7.12. SC_Filter_Data.....................................24
2.7.13. GF(256) Operations.................................25
3. FEC Packets...................................................25
3.1. Segmentation.............................................25
3.1.1. Transmit Blocks.....................................25
3.1.2. Working Blocks......................................26
3.1.3. Padding.............................................26
4. Parameter Selection...........................................26
5. Control Messages..............................................27
5.1. FEC Payload ID...........................................27
5.2. FEC Object Transmission Information......................27
5.2.1. FEC Encoding ID.....................................27
5.2.2. Common..............................................27
5.2.3. Scheme Specific.....................................28
6. Conventions used in this document.............................29
7. Security Considerations.......................................29
8. IANA Considerations...........................................29
9. References....................................................29
9.1. Normative References.....................................29
9.2. Informative References...................................29
10. Acknowledgments..............................................30
1. Introduction
This document describes a fully-specified FEC scheme for the
Supercharged forward error correction code. The Supercharged code is
designed for the erasure channel with performance very close to the
ideal Maximum Distance Separable(MDS) code and with very low
complexity. Section 2 describes the architecture of the code and
defines the generator matrices used by the code. Section 3 describes
how to construct FEC packets. Section 4 discusses code parameter
selection for a particular usage context. Section 5 defines the
protocol information elements. Section 6 considers security.
Section 7 considers IANA.
2. Supercharged Code
2.1.1. Definitions
ceil(a): rounds a to the nearest integer towards infinity
floor(a): rounds a to the nearest integer towards minus infinity
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min(a,b): returns the minimum of a and b
max(a,b): returns the maximum of a and b
a % b: is a modulo b
a + b: is a plus b
a * b: is a multiplied by b.
a ^ b: the bitwise XOR of a and b
a ^^ b: raises a to the b power
I_a: the a x a identity matrix
zeros(a,b): the a x b zero matrix
2.2. Overview
Figure 1 shows a general block diagram of the supercharged code. It
consists of a network of codes including block codes, repetition
codes, and parallel filter codes. Block code 1 consists of a
Vandermonde matrix in GF(256), a non-systematic Reed Solomon code.
Block code 2 and 3 consist of binary block codes.
+--------------+ +-----------------+
+---| Block Code 1 |---| Repetition Code |---+
| +--------------+ +-----------------+ |
| |
| +--------------+ +-----------------+ |
x ---+---| Block Code 2 |---| Repetition Code |---+----- y
| +--------------+ +-----------------+ |
| |
| +--------------+ +-----------------+ |
+---| Block Code 3 |---| | |
| +--------------+ | | |
| | Parallel Filter |---+
+----------------------| Code |
| |
+-----------------+
Figure 1 Block Diagram of the SC Code
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The parallel filter code of Figure 1 is detailed in Figure 2. It
consists of interleavers, tailbiting FIR filters, and a multiplexer
to select the output of the filters.
+----------------+
+---------------+ | Tailbiting FIR |
+---| Interleaver 1 |---| Filter |-------+
| +---------------+ | | |
| +----------------+ +-----+
---+ ... ... | Mux |---
| +----------------+ +-----+
| +---------------+ | Tailbiting FIR | |
+---| Interleaver M |---| Filter |-------+
+---------------+ | |
+----------------+
Figure 2 An example parallel filter code showing
individual data interleavers and tailbiting FIR filters as
coding components.
An example of one of the tailbiting FIR filters is illustrated in
Figure 3, where the state of the filter is initialized with the final
state to make it tailbiting.
+---+ +---+ +---+
---| D |---| D |---| D |
+---+ +---+ +---+
| | |
+-------+-------+
|
+--------------
Figure 3 An example 3 tap FIR filter that can be used for
the tailbiting FIR filter coding component. An XOR
operation is applied at the output of the delay elements
to produce the final output.
Optionally, if the number of transmit symbols N is signaled to be
limited such that N<=256, then the code can achieve ideal performance
by utilizing a Reed Solomon code.
2.3. Matrix Representation
Since supercharged codes are linear, an output codeword can be
expressed as a matrix multiplied by an input vector. Given Kx1
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encoding state vector x, consisting of binary transmit symbols, the
output Nx1 codeword, y, can be written as
y = (T*[I_K; B_3] + R_1*B_1 + R_2*B_2)*x (1)
where T is the N x (K+Num_B_3) generator matrix for the FIR
structure, B_1 is the Num_V_RS x K generator matrix for the first
block code, B_2 is the Num_B_2 x K generator for the second block
code, B_3 is the Num_B_3 x K generator matrix of the third block
code, and R_1 is a N x Num_V_RS stack and R_2 is a N x Num_B_2 stack
of identity matrices which facilitates repetition. For example,
matrix R_1 would consist of floor(N/Num_V_RS) copies of the identity
matrix stacked vertically, with a fractional identity matrix below
consisting of N mod Num_V_RS rows. The "+"operator indicates the
bitwise XOR operation. For convenience, denote the generator matrix
P = (T*[I_K; B_3] + R_1*B_1 + R_2*B_2), such that y=Px.
2.4. Systematic Encoding
Supercharged codes are not inherently systematic codes. Non-
systematic codes are commonly transformed into an effective
systematic code by pre-processing the input data before using it as
the input to the encoder, y=Px. The encoder input is calculated by
decoding the desired input data and running the decoder to determine
the encoder input vector x. Let matrix P_enc be the KxK generating
matrix corresponding to the first K elements of y, the encoder input
x can be computed using the following
x = P_enc^^(-1) * d.
Now, x can be used to encode using equation (1) to generate y. The
first K elements of vector y will be equal to d.
2.5. Erasure Channel
After encoding, the N transmit symbols of codeword vector y are
transmitted on the channel. Some of these transmit symbols are
erased by the channel. Suppose that the Nxr matrix E represents the
erasure pattern of the channel in that it selects out the r received
transmit symbols y_r from the transmitted symbols y. If the ith
received symbol is the jth transmit symbol, then E(i,j)=1. This
results in
y_r = E*y.
At the decoder, the effective generator matrix at the receiver is P_r
= E*P.
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2.6. Decoding
Decoding is the process of determining x given y_r and P_r. Decoding
can be implemented in several different ways, but each are equivalent
to solving the least squares problem x = (P_r^^T*P_r)^^-1 * P_r^^T *
y_r. Modern sparse matrix factorization techniques can take
advantage of the sparse structure imposed by the parallel filter
structure if (1) is rewritten in the following equivalent form
z = Gw, (2)
with augmented generator matrix G defined as
G = [ [B_3; B_2; B_1] I_L; T R_2 R_1]
and where the augmented output vector z=[zeros(L,1); y], the
augmented input vector w=[x; B_3*x; B_2*x; B_1*x], and where L=
Num_V_RS+Num_B_2+Num_B_3. The bottom L elements of vector w contain
the outputs, before repetition, of the block codes. These L values
are appended to vector x to form the augmented input vector w. The
first L rows of G implement the block code and XOR the block code
output with itself to generate the L zeros at the top of the z
vector. The subsequent N rows of G implement the FIR structure and
XOR the output with the output of the block codes.
This problem can be efficiently solved using direct sparse matrix
factorization techniques described in [3-8]. It is RECOMMENDED that
the Dulmage-Mendelsohn based solver in chapter 8 of [5] be used with
addition, multiplication, and division updated to support a finite
field. This algorithm utilizes pivoting based on node degrees in the
equivalent graph to minimize fill-in. The solution is completed by
performing forward and backward substitutions. Iterative solvers are
also possible.
Once the encoder state vector x, or equivalently the augmented
encoder state vector w, has been determined, the task remains to
determine the data vector d. For any elements of d that are missing,
then can be recovered by using appropriate rows of (1) or (2).
2.7. Matrix P Construction
2.7.1. Function Prototypes
The following functions are utilized to construction the Supercharged
code.
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[K_eff, Num_V_RS, Num_B_2, Num_B_3] = SC_Parameters(K, N)
K_eff=SC_K_table(K)
b=RNG(a)
a=RNG_2(a,b)
[permutation,the_seed]= Generate_Permutation(a,b)
G_V_RS = RS_gen(K,N)
[filter_data, filter_N]=SC_filter_data(z)
b=GF_exp(a)
C=GF_Multiply(A,B)
2.7.2. Parallel Filter Code T Construction
The parallel filter code matrix T can be generated using the
following pseudo code. The code generates multiple random
interleavers and selects which output of which interleaver depending
on the SID, where the SID is definded in section 3. Note that at
the receiver, only filter outputs corresponding to the received SID's
are required. The following code generates filter outputs for SIDs 0
to N-1. Determination of the filter output is a function of the SID
only, not any other filter output, making it simple to generate only
the filter outputs needed at encoding or decoding. The
Generate_Permutation function is defined in section 2.7.9. , the
SC_filter_data function is defined in section 2.7.12. , and the RNG
function is defined in section 2.7.8.
seed1 = 758492
seed2 = ( (K_eff*874) ^ (seed1) )
seed3 = 23091
base_permutation = Generate_Permutation(K_eff+Num_B_3,seed2)
filter_data = SC_filter_data(K_eff+Num_B_3)
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T = zeros(N,K_eff+NUM_B_3)
for SID=0:N-1
%Determine which filter to select
rn1 = min( RNG(15*(SID+1)+2*seed3) , 2^^32 )
index = 0
while(rn1>(filter_data[index]))
index = index+1
end
tdeg=index+1
%Determine which interleaver to select
rn2 = min( RNG(2*K_eff+3*(SID+1)) , 2^^32 )
interleaver_number = ( (rn2) % (K_eff+Num_B_3) )
%Determine which part of the interleaver to select
rn3 = min( RNG(98573+2*(SID+1)+rn1) , 2^^32 )
interleaver_part = ((rn3) % (K_eff+Num_B_3))
for tap_loop=0:tdeg
filter_tap = (tap_loop+interleaver_part) %
(K_eff+Num_B_3)
tap_location = (base_permutation[filter_tap] +
base_permutation[interleaver_number]) % (K_eff+Num_B_3)
T[Num_V_RS+Num_B_2+Num_B_3+SID,tap_location] = 1
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end
end
2.7.3. Repetition Code R Construction
The repetition code matrix R_1 and R_2 can be constructed via the
following pseudo code. Note that at the receiver, only filter
outputs corresponding to the received SID's are required. The
following code generates filter outputs for SIDs 0 to N-1 for R_1.
R_1 = zeros(N,Num_V_RS)
for SID = 0:N-1
for k = 0:Num_V_RS-1
if( ((SID-k) % (Num_V_RS)) == 0 )
R_1[SID,k] = 1
end
end
end
The following code generates filter outputs for SIDs 0 to N-1 for
R_2.
R_2 = zeros(N, Num_B_2)
for SID = 0:N-1
for k = 0: Num_B_2-1
if( ((SID-k) % (Num_B_2)) == 0 )
R_2[SID,k] = 1
end
end
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end
2.7.4. Block Code B_1 Construction
The Vandermonde matrix of block code B_1 can be constructed via the
following pseudo code. The GF_exp function is defined in section
2.7.13.
B_1 = zeros(Num_V_RS,K_eff)
for i = 0:Num_V_RS-1
for k = 0:K_eff-1
B_1[i+1,k+1] = GF_exp( ((i+1)*k) % (2^^8-1) )
end
end
2.7.5. Block Code B_2 and B_3 Construction
The block code B_2 and B_3 can be constructed jointly via the
following pseudo code, where B_23=[B_3; B_2].
B_23 = zeros(Num_B_2 + Num_B_3,K_eff)
for i = 0:K_eff-1
for k = 0: Num_B_2 + Num_B_3 - 1
if( ( (k-i) % (Num_B_2 + Num_B_3) ) == 0)
B_23[k,i] = 1
end
end
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end
m=1
for i = 0:K_eff-1
for k = 0: Num_B_2 + Num_B_3 - 1
if( ( (k-i-2*floor(m/( Num_B_2 + Num_B_3))) % (Num_B_2 +
Num_B_3) ) == 0)
B_23[k,i] = 1
end
m = m+1
end
end
m=2
for i = 0:K_eff-1
for k = 0: Num_B_2 + Num_B_3 - 1
if( ( (k-i-3*floor(m/( Num_B_2 + Num_B_3))) % (Num_B_2 +
Num_B_3) ) == 0)
B_23[k,i] = 1
end
m = m+1
end
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end
2.7.6. SC_Parameters
The following pseudo code determines a set of parameters needed for
matrix construction. The SC_K_table is defined in section 2.7.7.
function [K_eff, Num_V_RS, Num_B_2, Num_B_3] = SC_Parameters(K, N)
K_eff = SC_K_table(K)
Num_V_RS = 11 + floor(K_eff/10000)
Num_B = floor(K_eff^^(0.62)) + 3
if( K_eff >= 17376 )
Num_B = ceil( K_eff*0.0152 + 163 )
end
Num_B_3 = ceil(0.75*( Num_B ))
Num_B_2 = Num_B - Num_B_3
2.7.7. K Table
The function K_eff=SC_K_table(K) is implemented based on the
following table, by returning the smallest K_eff such that K_eff>=K.
10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,
33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,
56,57,58,59,60,61,62,63,64,65,66,67,69,70,71,72,73,74,75,76,77,78,79,
80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,1
02,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,11
9,120,121,122,123,124,125,126,127,128,129,130,131,133,134,135,136,137
,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,
155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,1
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72,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,18
9,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206
,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,
224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,2
41,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,25
8,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275
,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,
293,294,295,296,297,298,299,300,302,303,304,305,306,307,308,309,310,3
11,312,314,315,316,320,321,324,328,329,335,337,338,340,341,344,347,34
9,352,355,357,358,360,362,364,366,368,372,377,380,381,382,384,385,388
,389,393,394,395,397,399,405,408,409,410,411,416,418,424,426,428,431,
432,434,438,443,447,448,451,452,453,457,460,465,466,467,469,473,476,4
77,478,482,483,484,485,486,490,491,492,493,494,496,497,498,500,501,50
2,503,504,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520
,521,522,524,526,527,528,529,530,532,533,534,535,536,537,539,541,542,
543,545,546,549,551,552,553,554,555,557,558,559,561,562,563,564,566,5
69,571,572,573,574,576,577,578,579,580,582,583,585,586,587,588,589,59
0,592,593,594,597,598,599,600,602,603,606,607,608,609,610,612,614,615
,616,617,619,620,622,625,626,627,628,629,630,631,633,635,636,637,638,
640,643,645,648,650,652,653,654,655,656,659,660,661,662,664,666,667,6
68,669,672,673,674,675,677,687,688,691,692,693,694,695,696,698,699,70
0,701,703,710,711,712,715,716,717,718,726,727,730,731,734,736,737,741
,744,747,748,751,752,753,757,759,760,762,764,766,769,771,772,773,774,
775,777,778,779,786,788,790,792,793,794,795,797,798,799,800,801,802,8
04,805,810,811,812,813,815,820,821,822,823,825,827,829,830,831,834,83
5,837,838,839,840,843,844,845,846,848,849,851,852,853,854,857,858,860
,863,864,866,868,869,870,875,877,879,883,886,887,890,891,894,897,898,
899,900,902,903,904,905,906,907,909,912,913,914,917,922,926,927,928,9
31,934,938,940,942,944,945,948,950,953,954,960,961,963,967,968,970,97
1,972,974,977,979,980,981,985,987,989,990,995,996,1000,1002,1003,1005
,1006,1007,1009,1010,1015,1020,1021,1022,1024,1025,1027,1032,1033,103
4,1035,1037,1041,1042,1043,1046,1048,1050,1051,1054,1056,1057,1059,10
60,1062,1065,1069,1070,1071,1074,1076,1078,1079,1082,1083,1085,1086,1
087,1088,1089,1095,1098,1099,1106,1110,1111,1118,1120,1123,1124,1125,
1131,1132,1134,1136,1139,1140,1142,1144,1150,1152,1157,1161,1162,1165
,1169,1173,1175,1176,1179,1181,1182,1183,1194,1200,1201,1204,1205,120
6,1208,1209,1212,1213,1214,1218,1219,1220,1222,1225,1227,1228,1229,12
32,1236,1238,1240,1242,1243,1245,1248,1250,1252,1253,1255,1258,1261,1
269,1273,1278,1279,1280,1283,1284,1292,1293,1302,1303,1306,1310,1311,
1315,1318,1319,1321,1325,1330,1331,1342,1343,1347,1348,1352,1357,1359
,1361,1365,1374,1380,1382,1384,1388,1389,1390,1391,1392,1395,1397,140
3,1404,1407,1413,1417,1418,1420,1425,1429,1431,1435,1436,1437,1447,14
50,1461,1462,1464,1473,1474,1475,1477,1485,1490,1494,1496,1497,1502,1
503,1507,1513,1514,1516,1521,1522,1526,1530,1534,1539,1541,1549,1552,
1554,1555,1561,1564,1569,1572,1579,1585,1586,1590,1591,1593,1595,1596
,1597,1598,1600,1604,1608,1610,1611,1612,1616,1617,1624,1631,1633,163
6,1641,1646,1649,1650,1658,1660,1665,1667,1671,1673,1679,1683,1689,16
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92,1696,1698,1703,1705,1707,1708,1713,1716,1722,1728,1733,1734,1739,1
740,1742,1744,1745,1756,1759,1760,1764,1768,1771,1776,1777,1780,1782,
1787,1800,1807,1814,1824,1826,1827,1842,1844,1854,1857,1863,1867,1873
,1874,1878,1881,1883,1887,1889,1890,1891,1892,1894,1896,1903,1905,190
6,1910,1919,1924,1926,1931,1933,1943,1944,1948,1952,1954,1967,1971,19
73,1976,1979,1985,1986,1987,1989,1992,1994,1995,1998,2000,2005,2006,2
018,2019,2030,2040,2043,2048,2054,2055,2057,2061,2070,2071,2074,2077,
2082,2084,2087,2089,2093,2096,2098,2103,2104,2107,2111,2120,2122,2125
,2128,2138,2150,2152,2155,2160,2175,2177,2182,2189,2195,2200,2201,220
3,2217,2219,2225,2226,2231,2234,2235,2236,2237,2245,2247,2274,2276,22
78,2280,2282,2283,2286,2292,2303,2304,2306,2310,2315,2316,2319,2320,2
321,2330,2333,2336,2339,2343,2344,2345,2351,2367,2368,2371,2374,2382,
2389,2392,2395,2396,2400,2402,2407,2410,2412,2416,2421,2422,2434,2442
,2446,2447,2462,2473,2477,2478,2481,2486,2490,2492,2495,2502,2505,250
7,2509,2512,2513,2522,2525,2527,2528,2536,2543,2549,2556,2559,2561,25
63,2565,2583,2587,2590,2592,2596,2598,2601,2603,2604,2606,2617,2622,2
625,2626,2636,2638,2640,2643,2654,2660,2668,2673,2677,2679,2688,2695,
2699,2701,2713,2714,2723,2737,2741,2747,2753,2762,2764,2769,2772,2775
,2776,2785,2796,2802,2805,2808,2826,2828,2830,2831,2834,2836,2853,287
5,2877,2878,2884,2906,2938,2945,2948,2950,2961,2964,2966,2968,2979,29
80,2985,2989,2998,3008,3011,3015,3018,3022,3027,3048,3049,3051,3053,3
056,3062,3071,3075,3080,3093,3094,3095,3097,3101,3107,3109,3119,3122,
3128,3149,3150,3151,3158,3166,3167,3173,3178,3180,3181,3182,3186,3190
,3195,3200,3201,3203,3204,3205,3208,3216,3217,3223,3224,3232,3236,324
0,3248,3251,3253,3269,3276,3278,3279,3286,3292,3299,3306,3309,3336,33
40,3342,3344,3351,3352,3356,3357,3371,3375,3380,3387,3396,3404,3407,3
410,3423,3430,3445,3451,3463,3466,3471,3478,3479,3502,3513,3520,3528,
3531,3534,3539,3540,3546,3551,3565,3577,3579,3603,3606,3608,3612,3614
,3616,3620,3647,3650,3653,3658,3664,3677,3682,3686,3694,3697,3705,370
7,3724,3728,3744,3749,3751,3754,3761,3765,3776,3778,3781,3792,3797,37
99,3801,3834,3840,3841,3848,3861,3863,3883,3901,3903,3919,3924,3941,3
943,3960,3965,3970,3971,3989,3992,4007,4013,4015,4037,4039,4045,4050,
4055,4069,4072,4073,4091,4096,4106,4112,4124,4129,4133,4140,4146,4156
,4165,4188,4207,4209,4210,4215,4221,4236,4237,4247,4252,4253,4257,426
1,4266,4270,4318,4330,4341,4346,4359,4363,4365,4366,4388,4415,4418,44
36,4438,4453,4468,4474,4477,4503,4512,4513,4519,4522,4538,4548,4567,4
575,4576,4577,4583,4590,4621,4639,4651,4659,4681,4693,4698,4700,4702,
4729,4731,4739,4741,4742,4748,4749,4758,4764,4765,4771,4772,4780,4785
,4803,4804,4838,4840,4843,4868,4871,4878,4885,4898,4901,4918,4924,493
3,4939,4954,4959,4979,4982,4988,4991,4999,5000,5008,5021,5023,5030,50
39,5060,5062,5063,5096,5116,5137,5143,5145,5162,5163,5167,5172,5186,5
218,5225,5238,5240,5252,5260,5279,5285,5295,5301,5310,5314,5317,5331,
5332,5334,5348,5353,5354,5390,5391,5392,5405,5407,5432,5449,5451,5453
,5460,5464,5466,5471,5473,5477,5492,5506,5508,5537,5540,5543,5554,556
1,5566,5570,5576,5579,5587,5616,5637,5672,5674,5676,5684,5694,5716,57
32,5774,5792,5798,5800,5808,5823,5838,5844,5863,5896,5897,5899,5900,5
916,5921,5930,5960,5975,6039,6055,6057,6059,6067,6068,6078,6092,6099,
Stauffer, et al. Expires September 27, 2014 [Page 15]
Internet-Draft Supercharged Code March 2014
6102,6107,6136,6151,6169,6189,6191,6218,6233,6249,6271,6274,6296,6318
,6352,6363,6376,6407,6430,6435,6441,6463,6486,6491,6502,6512,6518,652
0,6534,6542,6549,6553,6589,6590,6593,6599,6614,6625,6634,6643,6655,66
70,6680,6684,6691,6692,6701,6708,6711,6724,6730,6732,6752,6799,6803,6
809,6812,6834,6849,6855,6877,6878,6879,6899,6907,6919,6936,6945,6946,
6954,6955,6956,6958,6981,7000,7011,7030,7032,7033,7108,7111,7127,7164
,7171,7175,7179,7181,7185,7225,7226,7281,7288,7295,7307,7325,7359,736
0,7390,7392,7411,7476,7520,7535,7548,7552,7558,7567,7589,7596,7616,76
45,7675,7679,7714,7726,7747,7770,7780,7785,7805,7818,7855,7870,7883,7
923,7935,7936,7953,7974,7999,8028,8030,8069,8074,8093,8104,8111,8122,
8150,8154,8172,8173,8189,8192,8193,8194,8223,8236,8290,8304,8377,8425
,8438,8439,8464,8481,8492,8521,8556,8559,8575,8582,8595,8602,8606,862
4,8628,8648,8654,8666,8672,8689,8738,8739,8744,8775,8787,8837,8841,88
42,8860,8928,8929,8970,8977,8993,9009,9019,9020,9029,9041,9051,9087,9
111,9151,9195,9208,9298,9303,9327,9344,9352,9360,9364,9388,9400,9402,
9446,9448,9449,9461,9462,9470,9485,9497,9512,9539,9546,9560,9572,9601
,9612,9642,9649,9653,9677,9689,9692,9704,9708,9758,9765,9794,9813,986
0,9916,9922,9927,9949,9971,9978,9981,9986,9987,10017,10040,10065,1007
3,10084,10097,10105,10120,10124,10134,10166,10187,10197,10202,10204,1
0241,10242,10279,10308,10324,10336,10351,10361,10458,10460,10567,1064
3,10676,10705,10712,10717,10759,10786,10787,10857,10883,10899,10911,1
0933,10944,10958,10963,11011,11015,11024,11036,11039,11049,11060,1111
9,11130,11146,11172,11203,11210,11216,11219,11230,11245,11316,11358,1
1371,11376,11423,11475,11534,11590,11649,11653,11677,11686,11707,1171
1,11740,11748,11751,11780,11823,11829,11843,11890,11896,11919,11947,1
1956,11976,12026,12037,12045,12072,12087,12108,12119,12154,12160,1220
8,12215,12216,12228,12229,12235,12247,12294,12333,12400,12437,12455,1
2458,12460,12469,12471,12510,12528,12567,12569,12593,12685,12694,1270
4,12721,12726,12754,12790,12817,12857,12914,12928,12936,12956,13002,1
3012,13026,13030,13035,13038,13057,13067,13082,13114,13143,13159,1319
3,13204,13214,13270,13278,13284,13326,13335,13417,13421,13423,13460,1
3479,13558,13607,13695,13696,13742,13764,13816,13827,13833,13837,1387
4,13879,13974,13987,14022,14100,14115,14140,14202,14272,14342,14350,1
4370,14376,14385,14393,14408,14409,14415,14417,14442,14486,14509,1456
0,14565,14713,14729,14743,14755,14798,14862,14874,14913,14934,14990,1
5007,15011,15120,15170,15194,15217,15227,15235,15285,15314,15321,1532
5,15332,15438,15499,15573,15611,15651,15668,15732,15735,15741,15757,1
5780,15808,15813,15847,15870,15941,15953,15977,16002,16017,16060,1610
8,16161,16286,16287,16304,16336,16374,16377,16384,16414,16505,16563,1
6623,16665,16670,16674,16689,16691,16710,16727,16743,16794,16828,1685
1,16900,16974,17005,17024,17029,17038,17039,17051,17086,17098,17148,1
7151,17195,17206,17266,17316,17323,17326,17331,17357,17376,17466,1748
9,17531,17559,17642,17681,17791,17868,17926,17929,17988,17991,18009,1
8026,18027,18056,18116,18168,18232,18307,18309,18438,18503,18504,1851
1,18590,18628,18629,18630,18636,18647,18672,18691,18694,18719,18909,1
8988,19023,19036,19096,19126,19132,19139,19193,19204,19210,19277,1930
4,19314,19325,19539,19544,19547,19631,19632,19635,19675,19700,19705,1
Stauffer, et al. Expires September 29, 2014 [Page 16]
Internet-Draft Supercharged Code March 2014
9740,19748,19921,19939,19951,19972,19985,20042,20052,20133,20141,2015
2,20173,20230,20245,20269,20287,20335,20355,20396,20407,20455,20501,2
0564,20580,20583,20664,20683,20710,20768,20776,20778,20789,20794,2098
8,21058,21087,21141,21143,21151,21186,21199,21216,21224,21385,21412,2
1468,21475,21478,21479,21486,21487,21515,21569,21616,21629,21673,2170
2,21729,21737,21747,21852,21927,21969,22060,22062,22068,22073,22114,2
2131,22244,22301,22320,22366,22433,22450,22482,22490,22498,22536,2272
7,22787,22947,22994,23010,23026,23063,23084,23135,23158,23180,23252,2
3392,23457,23491,23500,23568,23607,23721,23730,23787,23935,23971,2399
1,24023,24185,24215,24232,24398,24406,24476,24548,24550,24555,24562,2
4566,24591,24592,24616,24633,24673,24721,24735,24743,24761,24832,2489
1,24967,24976,25062,25080,25230,25391,25407,25433,25463,25493,25543,2
5613,25668,25756,25919,26022,26048,26050,26092,26291,26297,26329,2634
2,26371,26535,26566,26582,26676,26741,26838,26908,26910,26973,26984,2
7111,27119,27163,27256,27296,27353,27392,27428,27492,27594,27644,2766
6,27682,27771,27885,27895,27959,27987,28088,28116,28134,28137,28248,2
8263,28365,28466,28548,28549,28787,28816,28845,28966,29002,29042,2905
4,29072,29127,29138,29265,29326,29345,29434,29481,29487,29500,29588,2
9731,29816,29827,29868,29905,29964,30037,30097,30153,30169,30280,3034
6,30405,30433,30461,30493,30513,30550,30583,30646,30654,30909,30915,3
0921,30930,30974,30997,31052,31056,31142,31199,31283,31285,31303,3150
5,31578,31605,31948,31957,31997,32124,32139,32142,32272,32403,32555,3
2601,32630,32631,32648,32699,32768,32807,32849,32912,32932,32961,3296
5,33129,33171,33200,33282,33334,33623,34258,34302,34654,34708,35024,3
5031,35388,35395,35462,35488,35586,35600,35747,35750,35774,35802,3607
1,36112,36189,36252,36254,36294,36328,36357,36448,36476,36477,36479,3
6485,36637,36749,36849,36874,36894,37170,37185,37187,37227,37612,3769
5,37701,37767,37793,37805,37815,37826,37906,37992,38008,38010,38046,3
8080,38130,38236,38385,38763,38787,39166,39176,39201,39237,39288,3939
8,39482,39643,39786,39831,39960,39980,40089,40105,40140,40152,40192,4
0220,40274,40293,40303,40398,40549,40604,40625,40666,40690,40816,4084
3,40847,40894,40896,40962,40969,41003,41087,41107,41132,41216,41226,4
1265,41314,41321,41357,41367,41539,41576,41641,41717,41820,42033,4206
7,42172,42490,42662,42795,42813,42916,43339,43351,43388,43482,43498,4
3691,43840,43905,43924,43932,44033,44129,44279,44821,44883,44945,4495
1,45097,45162,45359,45389,45557,45582,45638,45813,45830,45919,45960,4
6038,46086,46104,46187,46281,46428,46463,46481,46574,47047,47324,4741
8,47523,47717,48007,48264,48334,48489,48501,48702,48788,48976,48994,4
9504,49550,49703,49711,49978,49995,50006,50338,50511,50799,50946,5094
7,50951,50980,51017,51150,51244,51530,51616,51977,52007,52062,52364,5
2441,52586,52598,52768,52883,52978,53047,53064,53114,53127,54024,5454
6,54578,54735,54803,55123,55289,55510,55661,55744,55843,55885,55921,5
6297,56403,56696,57113,57424,57614,57779,58294,58326,58721,58908,5934
6,59541,59651,59882,60076,60164,60250,60618,60799,61144,61208,61217,6
1617
Stauffer, et al. Expires September 29, 2014 [Page 17]
Internet-Draft Supercharged Code March 2014
2.7.8. Random Number Generator
The SC code utilizes two random number generators. The first uses
the second. The first is described by the following pseudo code:
function b=RNG(a)
for i = 0:7
a = RNG_2( a, ( (a) % (89) ) )
b = (b) % (a)
end
The second random number generator uses a selectable set of feedback
taps. The second is described by the following pseudo code:
function a=RNG_2(a,b)
tap_list=[32, 31, 30, 10
32, 31, 29, 1
32, 31, 26, 18
32, 31, 26, 9
32, 31, 26, 7
32, 31, 23, 10
32, 31, 22, 17
32, 31, 21, 16
32, 31, 21, 5
32, 31, 18, 10
32, 31, 16, 2
32, 31, 15, 10
32, 31, 14, 4
Stauffer, et al. Expires September 29, 2014 [Page 18]
Internet-Draft Supercharged Code March 2014
32, 31, 13, 8
32, 31, 9, 7
32, 31, 5, 4
32, 30, 29, 23
32, 30, 29, 20
32, 30, 29, 16
32, 30, 29, 15
32, 30, 27, 24
32, 30, 27, 21
32, 30, 27, 12
32, 30, 27, 8
32, 30, 26, 25
32, 30, 26, 13
32, 30, 25, 16
32, 30, 23, 16
32, 30, 23, 14
32, 30, 23, 4
32, 30, 21, 14
32, 30, 19, 8
32, 30, 19, 4
32, 30, 17, 3
32, 30, 15, 6
32, 30, 11, 8
32, 30, 11, 5
Stauffer, et al. Expires September 29, 2014 [Page 19]
Internet-Draft Supercharged Code March 2014
32, 30, 8, 3
32, 30, 7, 4
32, 29, 28, 19
32, 29, 27, 23
32, 29, 27, 21
32, 29, 27, 6
32, 29, 26, 6
32, 29, 25, 6
32, 29, 22, 18
32, 29, 19, 16
32, 29, 17, 15
32, 29, 15, 8
32, 29, 6, 5
32, 29, 6, 4
32, 28, 25, 15
32, 28, 25, 11
32, 28, 25, 6
32, 28, 23, 6
32, 28, 15, 13
32, 28, 9, 7
32, 27, 26, 14
32, 27, 25, 20
32, 27, 25, 19
32, 27, 25, 17
Stauffer, et al. Expires September 29, 2014 [Page 20]
Internet-Draft Supercharged Code March 2014
32, 27, 25, 7
32, 27, 25, 5
32, 27, 23, 6
32, 27, 21, 6
32, 27, 20, 18
32, 27, 18, 14
32, 27, 15, 14
32, 27, 14, 12
32, 27, 14, 9
32, 27, 8, 6
32, 26, 25, 10
32, 26, 23, 12
32, 26, 22, 7
32, 26, 20, 11
32, 26, 19, 9
32, 26, 19, 7
32, 26, 18, 13
32, 26, 15, 7
32, 25, 24, 7
32, 25, 22, 15
32, 25, 17, 7
32, 25, 14, 13
32, 24, 22, 13
32, 23, 21, 16
Stauffer, et al. Expires September 29, 2014 [Page 21]
Internet-Draft Supercharged Code March 2014
32, 23, 18, 14
32, 21, 20, 19
32, 20, 17, 15
32, 19, 18, 13]
taps[0]=tap_list[b,0]
taps[1]=tap_list[b,1]
taps[2]=tap_list[b,2]
taps[3]=tap_list[b,3]
feedback=2.^^(32-taps[0]) + 2.^^(32-taps[1]) + 2.^^(32-taps[2]) +
2.^^(32-taps[3])
if( (a) & (1) )
a = (a) ^ (feedback)
a = (a) >> (1)
a = (2^31) || (a)
else
a = (a) >> (1)
end
2.7.9. Random Permutation
The SC code utilizes a random permutation of length K to facilitate
the construction of the random interleavers needed for the parallel
filter codes. The random permutation is given by the following
pseduocode. The RNG_2 function is defined in section 2.7.8.
function [permutation,the_seed]= Generate_Permutation(a,b)
for i=0:a-1
permutation[i] = i + 1
Stauffer, et al. Expires September 29, 2014 [Page 22]
Internet-Draft Supercharged Code March 2014
end
for i=0:a-1
c = RNG_2(b,1)
b = ( (c) % (a-(i-1)) ) + i
d = permutation[i]
permutation[i] = permutation[b]
permutation[b] = d
end
2.7.10. RS Generator
A Reed Solomon code is utilized in the construction of the SC code.
Its construction is described by the following pseudo code. The
GF_exp and the GF_Multiply functions are defined in section 2.7.13.
function G_V_RS = RS_gen(K,N)
Gt=zeros[N,K]
for i=0:N-1
for k=0:K-1
a = ((i+1)*k) % (2^^8-1)
Gt[i,k]=GF_exp(a)
end
end
G1=Gt[1:K,1:K]
G2=Gt[K+1:N,1:K]
Stauffer, et al. Expires September 29, 2014 [Page 23]
Internet-Draft Supercharged Code March 2014
G_V_RS = GF_Multiply(G2,G1^^-1)
GF_Multiply implementes G2*G1_inv where the multiplication and
addition are performend in the GF field. The matrix inverse G1^^-1
can be easily implemented using Gaussian Elimination for the small
matrix G1.
2.7.11. RS Code
If the number of transmit symbols N is optionally limited to N<=256
and signaled using R=1, then the following pseudo code is used to
generate matrix P. The RS_gen function is defined in section 2.7.10.
N is given by the FEC-OTI-Max-Number-of-Encoding-Symbols.
Num_V_RS = N - K
B_1 = RS_gen(K,K+Num_V_RS)
P = [I[K]
B_1 ]
Num_B = 0
K_eff = K
2.7.12. SC_Filter_Data
[Filter_data, filter_N]=SC_filter_data(z)
Filter_data=[0,2147483648,2863311531,3221225472,3435973837,3579139413
,3681400539,3758096384,3817748708,3865470566,3904515724,3937053355,39
64585196,3988183918,4008636143,4026531840,4042322161,4056358002,40689
16386,4080218931,4090445044,4099741510,4108229587,4116010325,41231686
04,4129776246,4135894433,4141575607,4146864975,4151801719,4156419964,
4160749568,4164816772,4168644728,4172253945,4175662649,4178887099,418
1941841,4184839929,4187593114,4190211996,4192706170,4195084336,419735
4403,4199523578,4201598442,4203585013,4205488811,4207314902,420906795
0,4210752251,4212371771,4213930177,4215430865,4216876982,4218271451,4
219616993,4220916136,4222171240,4223384508,4224557996,4225693630,4226
793212,4227858432,4228890876,4229892034,4230863307,4231806012,4232721
Stauffer, et al. Expires September 29, 2014 [Page 24]
Internet-Draft Supercharged Code March 2014
393,4233610620,4234474799,4235314972,4236132128,4236927197,4237701065
,4238454568,4239188500,4239903613,4240600621,4241280205,4241943008,42
42589646,4243220702,4243836733,4244438269,4245025816,4245599856,42461
60849,4246709236,4247245437,4247769853,4248282869,4248784852,42492761
55,4249757114,4250228053,4250689283,4251141099,4251583788,4294967295]
filter_N=min(100,z)
Filter_data[Filter_N-1]=4294967295
2.7.13. GF(256) Operations
The SC code utilizes Galois field arithmetic in GF(256). The
primitive polynomial is D^^8 + D^^4 + D^^3 + D^^2 + 1. The
b=GF_exp(a) function raises the primitive element to the supplied
power, a. The function C=GF_Multiply(A,B) multiplies two matrices in
the Galois field.
3. FEC Packets
Encoded packets are constructed using a 4 byte FEC Payload ID
followed by transmit symbols. The Source ID field (SID) of the FEC
Payload ID identifies the Source ID of the first transmit symbol in
the packet. Subsequent transmit symbols have sequential increasing
SIDs. If the last transmit symbol of a packet contains source
padding, these padding bytes may be excluded from the packet.
Otherwise, packets must contain only whole transmit symbols.
It is RECOMMENDED that each packet include exactly one transmit
symbol. Multiple transmit symbols per packet SHALL also be
supported.
3.1. Segmentation
In order to encode large files within the working memory constraint,
the source file may need to be segmented into transmit blocks and
working blocks.
3.1.1. Transmit Blocks
Given a source file of size F bytes and a transmit symbol size of T
bytes, the file can be divided into K_total=ceil(F/T) transmit
symbols. A source transmit block is a collection of KL or KS of
these transmit symbols. KL and KS may be different if the total
number of source transmit blocks does not evenly divide the number of
transmit symbols required to represent the file. The number of
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source transmit blocks with KL transmit symbols and the number of
source transmit blocks with KS transmit symbols are communicated to
the decoder using parameter Z. After encoding, a transmit block
consists of a source transmit block and a repair transmit block.
The transmit blocks are ordered such that the first ZL transmit block
are encoded from source transmit blocks of size KL transmit symbols.
The remaining ZS transmit blocks are encoded from source transmit
blocks are of size KS transmit symbols. Given Z, the first
ZL=ceil(K_total/Z)*Z-K_total transmit blocks are of size
KL=floor(K_total/Z) and the remaining ZS=K_total-floor(K_total/Z)*Z
transmit blocks are of size KS=ceil(K_total/Z).
3.1.2. Working Blocks
In order to satisfy the working memory requirement, the transmit
symbols can be further subdivided into working symbols. The working
symbols are ordered in a packet such that the first ceil(T/AL/Ns)*Ns-
T/AL working-blocks are of size TWL=floor(T/AL/Ns) and the remaining
T/AL-floor(T/AL/Ns)*Ns working-blocks are of size TWS=ceil(T/AL/Ns)
in a given packet. A working block is then a collection of working
symbols. The size of the working symbols are selected such that an
entire source working block can fit into the working memory, where
the source working block is the portion of the working block
consisting of only source data and not repair data. The ith working
block consists of the ith working symbol of transmit symbols of a
transmit block. The KL (or KS) transmit symbols of a source transmit
block can be viewed as a collection of working symbols or
equivalently as a collection of source working blocks.
After encoding, a working block consists of a source working block
and a repair working block. The receiver attempts to decode on a
subset of the source and repair working symbols in a working block.
3.1.3. Padding
In cases where effective number of transmit symbols used by the
encoder and decoder, K_eff, is K_eff>K, then K_eff-K transmit symbols
must be padded (with 0) to the data before encoding. These padded
symbols do not need to be transmitted, as the decoder is aware that
they are padding. (Padding SIDs 0 to K_eff-K-1 MAY be transmitted,
but it is RECOMMENDED that they are not.)
4. Parameter Selection
The code requires F, T, Z, Ns, and AL. F is the total file size in
Bytes. T is the transmit symbol size in bytes, and it is RECOMMENDED
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that it be equal to the packet payload size. The number of transmit
blocks Z MUST be chosen such that KL<=K_max, where KL is computed in
section 3.1.1. K_max is the maximum value in section 2.7.7.
The number of working symbols, Ns, MUST be chosen small enough such
that KL*TWL is less than or equal to the working memory requirement.
The byte alignment, AL, is to be chosen based on the protocol and the
typical machine architectures, a value of 4 (bytes) is RECOMMENDED.
5. Control Messages
This section describes control messages that are used by the FEC.
All fields are big-endian.
5.1. FEC Payload ID
The FEC payload ID is a 4-byte field defined as follows:
[0:7] TBN, (8 bits, unsigned integer): A non-negative integer
identifier indicating the transmit block number.
[8:31] SID , (24 bits, unsigned integer): A non-negative integer
identifier indicating the transmit symbols in the packet. SID 0 to
K-1 indicate systematic symbols.
The FEC Payload ID is shown in Figure 4.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| TBN | SID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 4 FEC Payload ID format
5.2. FEC Object Transmission Information
5.2.1. FEC Encoding ID
The value of the FEC Encoding ID MUST be 7, as assigned by IANA (see
Section 8).
5.2.2. Common
The Common FEC Object Transmission Information elements used by this
FEC Scheme are:
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[0:39] Transfer Length (F), (40 bits, unsigned integer): A non-
negative integer. This is the transfer length of the object in
bytes.
[40:47] are reserved.
[48:63] Transmit Symbol Size (T), (16 bits, unsigned integer): A
positive integer that is less than 2^^16. This is the size of a
transmit symbol in units of bytes.
The encoded Common FEC Object Transmission Information format is
shown in Figure 5.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Transfer Length (F) |
+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | Reserved | Symbol Size (T) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 5 Encoded Common FEC Object Transmission
Information for Supercharged FEC Scheme
5.2.3. Scheme Specific
The following parameters are carried in the Scheme-Specific FEC
Object Transmission Information element for this FEC Scheme:
[0:7] Z: The number of transmit blocks (8 bits, unsigned integer)
[8:23] Ns: The number of working blocks (16 bits, unsigned integer)
[24:30] AL: A symbol alignment parameter (7 bits, unsigned integer)
[31] R: 0: Default 1: OPTIONALLY indicates that the maximum value of
N satisfies N<=256 (1 bit, boolean)
The encoded Specific FEC Object Transmission Information format is
shown in Figure 5.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Z | Ns | Al |R|
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 6 FEC Payload ID format
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6. Conventions used in this document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC-2119 [RFC2119].
In this document, these words will appear with that interpretation
only when in ALL CAPS. Lower case uses of these words are not to be
interpreted as carrying RFC-2119 significance.
7. Security Considerations
Users could potentially be subject to a denial of service attack if a
single erroneous packet is injected into the delivery stream.
Therefore, it is RECOMMENDED that source authentication and integrity
checking are applied to the file or data object before delivering
decoded data to applications. The hashing methodology of SHA-256 is
an example [2].
8. IANA Considerations
Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
registration. For general guidelines on IANA considerations as they
apply to this document, see [RFC5052]. IANA is requested to assign a
value under the ietf:rmt:fec:encoding name-space to "Supercharged
Code" as the FEC Encoding ID value associated with this
specification, preferably the value 7.
9. References
9.1. Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels", BCP 14, RFC 2119, March 1997.
[2] "Secure Hash Standard", National Institute of Standards
and Technology FIPS PUB 180-3, October 2008.
9.2. Informative References
[3] Timothy Vismor, "Matrix Algorithms."
[4] Sergio Pissanetzky, "Sparse Matrix Technology," Academic Press,
London (1984).
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[5] Timothy A. Davis, "Direct Methods for Sparse Linear Systems"
SIAM, Philadelphia, Pa (2006)
[6] Yousef Saad, "Iterative Methods for Sparse Linear Systems" 2nd
Ed. SIAM, Philadelphia, Pa (2003)
[7] I.S. Duff, A.M. Erisman, and J. K. Reid, "Direct Methods for
Sparse Matrices" (2008) (ISBN: 978-0198534082)
[8] John K. Reid, "Solution of linear systems of equations: Direct
methods" (1977)
[9] Golub, G.H. "Numerical methods for solving linear least-squares
problems" Numerische Mathematik Volumne 7, Number 3 (1965) pp
206-216
10. Acknowledgments
This document was prepared using 2-Word-v2.0.template.dot.
Authors' Addresses
Erik Stauffer
Broadcom
190 Mathilda Place
Sunnyvale, Ca 94086
Email: eriks@broadcom.com
BZ Shen
Broadcom
5300 California Avenue
Irvine, CA 92617
Email: bzshen@broadcom.com
Soumen Chakraborty
Broadcom
RMZ Ecospace
Bellandur
Bangalore 560037, India
Email: soumen@broadcom.com
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Djordje Tujkovic
Broadcom
190 Mathilda Place
Sunnyvale, Ca 94086
Email: djordje@broadcom.com
Jing Huang
Broadcom
190 Mathilda Place
Sunnyvale, Ca 94086
Email: jingh@broadcom.com
Shiv Shet
Broadcom
RMZ Ecospace
Bellandur
Bangalore 560037, India
Email: shivaprakash@broadcom.com
Kamlesh Rath
Broadcom
190 Mathilda Place
Sunnyvale, Ca 94086
Email: krath@broadcom.com
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