RMT V. Roca Internet-Draft C. Neumann Expires: December 30, 2005 INRIA D. Furodet STMicroelectronics June 28, 2005 Low Density Parity Check (LDPC) Forward Error Correction draft-roca-rmt-ldpc-00.txt Status of this Memo By submitting this Internet-Draft, each author represents that any applicable patent or other IPR claims of which he or she is aware have been or will be disclosed, and any of which he or she becomes aware will be disclosed, in accordance with Section 6 of BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet- Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. This Internet-Draft will expire on December 30, 2005. Copyright Notice Copyright (C) The Internet Society (2005). Abstract This document describes an Under-Specified FEC Scheme that can be used with the broad class of Low Density Parity Check (LDPC) codes and their application to the reliable delivery of objects on packet erasure channels. Additionally, this document describes the LDPC- Staircase and LDPC-Triangle Forward Error Correction codes, two instances of the LDPC FEC Scheme, in a way that enables fully inter- Roca, et al. Expires December 30, 2005 [Page 1] Internet-Draft LDPC Forward Error Correction June 2005 operable implementations. The LDPC codes belong to the class of large block FEC codes, as defined in RFC3453, which enables them to efficiently encode/decode large objects, in a single block. They also enable a receiver to recover the k source symbols from any set of a little bit more than k encoding symbols. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Requirements notation . . . . . . . . . . . . . . . . . . . . 4 3. Definitions, Notations and Abbreviations . . . . . . . . . . . 5 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Abbreviations . . . . . . . . . . . . . . . . . . . . . . 6 4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 7 4.1 FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 7 4.2 FEC Object Transmission Information . . . . . . . . . . . 7 4.2.1 Mandatory Elements . . . . . . . . . . . . . . . . . . 7 4.2.2 Common Elements . . . . . . . . . . . . . . . . . . . 7 4.2.3 Scheme-Specific Elements . . . . . . . . . . . . . . . 8 4.2.4 Encoding Format . . . . . . . . . . . . . . . . . . . 8 5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.2 Parity Check Matrix . . . . . . . . . . . . . . . . . . . 10 5.3 Derivations and Interpretation of the Fields Provided in the FPI and FEC OTI . . . . . . . . . . . . . . . . . . 10 5.4 Pseudo Random Number Generator . . . . . . . . . . . . . . 11 6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 12 6.1 Instance Specific Parameters . . . . . . . . . . . . . . . 12 6.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 12 6.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 14 7. Full Specification of the LDPC-Triangle Scheme . . . . . . . . 15 7.1 Instance Specific Parameters . . . . . . . . . . . . . . . 15 7.2 Parity Check Matrix Creation . . . . . . . . . . . . . . . 15 7.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 16 8. Security Considerations . . . . . . . . . . . . . . . . . . . 17 9. Intellectual Property . . . . . . . . . . . . . . . . . . . . 18 10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 19 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 20 11.1 Normative References . . . . . . . . . . . . . . . . . . . 20 11.2 Informative References . . . . . . . . . . . . . . . . . . 20 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 21 A. Iterative Decoding Algorithm (Informative) . . . . . . . . . . 22 Intellectual Property and Copyright Statements . . . . . . . . 24 Roca, et al. Expires December 30, 2005 [Page 2] Internet-Draft LDPC Forward Error Correction June 2005 1. Introduction RFC 3453 [RFC3453] introduces large block FEC codes as an alternative to small block FEC codes like Reed-Solomon. The main advantage of such large block codes is the possibility to operate efficiently on source blocks of several tens of thousands (or more) source symbols of size. The present document introduces the Under-Specified FEC Encoding ID 132 that is intended to be used with the "Low Density Parity Check" (LDPC) FEC codes, that belong the class of large block codes. LDPC codes rely on a dedicated matrix, called a "Parity Check Matrix", at the encoding and decoding ends. The parity check matrix defines relationships (or constraints) between the various encoding symbols (i.e. source symbols and repair symbols), that are later used by the decoder to reconstruct the original k source symbols if some of them are missing. These codes are systematic, in the sense that the encoding symbols include the source symbols in addition to the redundant symbols. -- editor's note: This document makes use of the FEC Encoding ID value 132, but this may change after IANA assignment -- Since the encoder and decoder must operate on the same parity check matrix, some information must be communicated between them, as part of the FEC Object Transmission Information. Its content and the associated EXT_FTI are fully described in Section 4.2. The two variants specified in this document belong to this broad class of LDPC codes. But other codes, existing or forthcoming, may also be added in the future, taking advantage of the framework provided by the Under-Specified FEC Encoding ID 132. More specifically, this document reserves the FEC Instance ID value 0 for the LDPC-Staircase codes [Roca04][Mac03] and reserves the FEC Instance ID value 1 for the LDPC-Triangle codes [Roca04]. A publicly available reference implementation of these codes is available and distributed under a GNU/LGPL license [LDPCrefimpl]. Roca, et al. Expires December 30, 2005 [Page 3] Internet-Draft LDPC Forward Error Correction June 2005 2. Requirements notation 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 [RFC2119]. Roca, et al. Expires December 30, 2005 [Page 4] Internet-Draft LDPC Forward Error Correction June 2005 3. Definitions, Notations and Abbreviations 3.1 Definitions This document uses the same terms and definitions as those specified in [fec-bb-revised]. Additionally, it uses the following definitions: Encoding Symbol Group: a group of encoding symbols that are sent together, within the same packet, and whose relationships to the source object can be derived from a single Encoding Symbol ID. Source Packet a data packet containing only source symbols. Repair Packet a data packet containing only repair symbols. 3.2 Notations This document uses the following notations: L denotes the object transfer length in bytes k denotes the number of source symbols in a source block n denotes the number of encoding symbols E denotes the encoding symbol length in bytes B denotes the maximum source block length in terms of symbols N denotes the number of source blocks into which the object shall be partitioned G denotes the number of encoding symbols per group, i.e. the number of symbols sent in the same packet rate denotes the so-called "code rate", i.e. the k/n ratio max_n Maximum Number of Encoding Symbols per encoding block. This depends on FEC code rate. rand(m) denotes a pseudo-random number generator, that returns a new random integer in [0; m-1] each time it is called. Roca, et al. Expires December 30, 2005 [Page 5] Internet-Draft LDPC Forward Error Correction June 2005 3.3 Abbreviations This document uses the following abbreviations: ESI Encoding Symbol ID Roca, et al. Expires December 30, 2005 [Page 6] Internet-Draft LDPC Forward Error Correction June 2005 4. Formats and Codes 4.1 FEC Payload IDs The FEC Payload ID is composed of the Source Block Number and the Encoding Symbol ID: The Source Block Number identifies from which source block of the object the encoding symbol(s) in the payload is(are) generated. The Encoding Symbol ID identifies which specific encoding symbol generated from the source block is carried in the packet payload. Each encoding symbol is either an original source symbol or a redundant symbol generated by the encoder. There MUST be exactly one FEC Payload ID per packet. When multiple encoding symbols are sent in the same packet, the FEC Payload ID refers to the first symbol of the packet. The other symbols can be deduced as explained in Section 5.1 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 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Source Block Number | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Encoding Symbol ID | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 1: FEC Payload ID encoding format for FEC Encoding ID 132 4.2 FEC Object Transmission Information 4.2.1 Mandatory Elements FEC Encoding ID: the Under-Specified FEC Scheme described in this document uses the FEC Encoding ID 132. FEC Instance ID: this document reserves the FEC Instance ID value 0 for the LDPC-Staircase codes (Section 6) and the FEC Instance ID value 1 for the LDPC-Triangle codes (Section 7). 4.2.2 Common Elements The following elements MUST be used with the present FEC Scheme: Transfer-Length: a non-negative integer indicating the length of the object in bytes. Roca, et al. Expires December 30, 2005 [Page 7] Internet-Draft LDPC Forward Error Correction June 2005 Encoding-Symbol-Length: a non-negative integer indicating the length of each encoding symbol in bytes. Maximum-Source-Block-Length: a non-negative integer indicating the maximum number of source symbols in a source block. Max-Number-of-Encoding-Symbols: a non-negative integer indicating the maximum number of encoding symbols (i.e. source plus repair symbols in the case of a systematic code). Section 5.3 describes how to derive the values of each of these elements. 4.2.3 Scheme-Specific Elements PRNG seed: Seed (a 32 bit value) used to initiate the Pseudo Random Generator (defined in Section 5.4). This element is optional and may be used by some specific Instance IDs. Other elements MAY be defined for Instance-Specific needs. 4.2.4 Encoding Format This section shows possible encoding formats of the above FEC OTI. 4.2.4.1 Using the General EXT_FTI Format 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 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | HET = 64 | HEL | | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ + | Transfer-Length (L) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | FEC Instance ID | Encoding Symbol Length (E) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Max Source Block Length (B) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Max Nb of Enc. Symbols (max_n) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ .. Scheme Specific optional elements . +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Roca, et al. Expires December 30, 2005 [Page 8] Internet-Draft LDPC Forward Error Correction June 2005 5. Procedures This section defines procedures that are common to all FEC Instance IDs scoped by FEC Encoding ID 132. 5.1 General The source object is first partitioned into blocks, using the block partitioning algorithm specified in [fec-bb-revised]. To that purpose, the B (maximum source block length in symbols), L (object transfer length in bytes), and E (encoding symbol length in bytes) arguments are provided. As an output, the object is partitioned into N source blocks. These blocks are numbered consecutively from 0 to N-1. The first I source blocks consist of A_large source symbols, the remaining N-I source blocks consist of A_small source symbols. Each source symbol is E bytes in length, except perhaps the last symbol which may be shorter as explained in [fec-bb-revised]. FEC encoding and decoding is done block per block, independently. When multiple encoding symbols are sent in the same packet, it MUST be possible to identify each symbol from this single FEC Payload ID. To that purpose, the symbols of an Encoding Symbol Group (i.e. packet): o MUST be in sequence, from ESI i to ESI i+G-1 (inclusive), o MUST all be either source symbols, or repair symbols. Therefore, only source packets and repair packets are permitted, not mixed ones. The FEC Payload ID information MUST refer to the first encoding symbol of the packet. This specification does not specify what value for B should be used. This decision SHOULD be clarified either at implementation time, when the target use case is known, or in the specification of a FEC Instance ID, for instance to take into account some specificities of a FEC scheme. Similarly, this specification does not specify if and when Encoding Symbol Groups should be used or not, i.e. if and when we have G>1. This decision SHOULD be clarified either at implementation time, when the target use case is known, or in the specification of a FEC Instance ID, for instance to take into account some specificities of a FEC scheme. In both cases, a receiver can derive the B and G values from the Roca, et al. Expires December 30, 2005 [Page 9] Internet-Draft LDPC Forward Error Correction June 2005 information it receives. 5.2 Parity Check Matrix LDPC codes rely on a parity check matrix, which represents a linear equation system between repair symbols and source symbols of a given block. The basic operator is XOR and the matrix can only be filled with 1s and 0s. The parity check matrix is logically divided into two parts: the left side (from column 0 to k-1) which describes the occurrence of each source symbol in the equation system; and the right side (from column k to n-1) which describes the occurrence of each repair symbol in the equation system. An entry (a "1") in the matrix at position (i,j), i.e. at row i and column j, means that the symbol with ESI i appears in equation j. 5.3 Derivations and Interpretation of the Fields Provided in the FPI and FEC OTI The fields provided in the FEC OTI are derived using the "n-algorithm", described below: AT A SENDER: Input: B Maximum Source Block Length, i.e., the maximum number of source symbols per source block. This is given by the FEC codec specifications and/or the execution environment limitations. k Source Block Length, i.e., the number of source symbols per source block. This is given by source blocking algorithm. rate or (k,n) FEC code rate, which is given by the user (e.g. when starting a FLUTE sending application). It is expressed either as a floating point value, R, or as a quotient k/n. The latter option is RECOMMENDED for the integer math version of the algorithm. Output: max_n Maximum Number of Encoding Symbols per encoding block. This depends on FEC code rate. n Encoding Block Length, i.e., the number of encoding symbols generated for the source block. Roca, et al. Expires December 30, 2005 [Page 10] Internet-Draft LDPC Forward Error Correction June 2005 Algorithm: a. max_n = B / R rounded down to the nearest integer (max_n = (B * b) div a) b. n = k * max_n / B rounded down to the nearest integer (n = (k * max_n) div B) AT A RECEIVER: Input: B, max_n, k Output: n Algorithm: a. n = k * max_n / B rounded down to the nearest integer (n = (k * max_n) div B) Notes: (1) X div Y denotes the integer quotient of the division X/Y The use of floating point arithmetic in the algorithm might lead to erroneous results caused by rounding problems, depending on the mathematical library used. These problems can be avoided by using only integer math in all algorithm calculations. It is strongly recommended not to use rounding functions, and how to do that is presented in brackets 5.4 Pseudo Random Number Generator The present FEC Encoding ID relies on a pseudo-random number generator that must be fully specified in order to enable the receivers and the senders to build the same parity check matrix. -- editor's note: The PRNG to use is TBD. Current implementation relies on the GNU C Library lrand48() function, but this may not be the most appropriate choice. -- Roca, et al. Expires December 30, 2005 [Page 11] Internet-Draft LDPC Forward Error Correction June 2005 6. Full Specification of the LDPC-Staircase Scheme 6.1 Instance Specific Parameters LDPC-Staircase is identified by th Under-Specified FEC Encoding ID 132 and the the FEC Instance ID 0. LDPC-Staircase is based on a pseudo-random number generator as specified in Section 5.4. Therefore the seed used to initiate the PRNG is an instance-specific FEC Object Transmission Information element and MUST be transmitted within the FEC OTI, as specified in Section 4.2. 6.2 Parity Check Matrix Creation The matrix creation algorithm for LDPC Staircase is described in the following. The algorithm can be divided into two parts: The left side of the matrix where the occurrence of the source symbols in the equations is described, and the right side of the matrix where repair symbols are described. The left side is generated with the following algorithm: Roca, et al. Expires December 30, 2005 [Page 12] Internet-Draft LDPC Forward Error Correction June 2005 /* initialize a list of possible choices to * guarantee a homogeneous "1" distribution */ for(h = 3*k-1; h >= 0; h--) { u[h] = h % (n-k); } /* left limit within the list of possible choices, u[] */ t = 0; for(j = 0; j < k; j++) { /* for each source symbol column */ for(h = 0; h < 3; h++) { /* add 3 "1s" */ /* check that valid available choices remain */ for(i = t; i < 3*k && matrix_has_entry(u[i],j); i++); if(i < 3*k) { /* choose one index within the * list of possible choices */ do { i = t + rand() % (3*k-t); } while (matrix_has_entry(u[i],j)); matrix_insert_entry(u[i],j); /* replace with u[t] which has never been chosen */ u[i] = u[t]; t++; } else { /* no choice left, choose one randomly */ do { i = rand() % (n-k); } while (matrix_has_entry(i,j)); matrix_insert_entry(i,j); } } } /* Add extra bits to avoid rows with less than two checks. */ for(i = 0; i < n-k; i++) { /* for each row */ if(degree_of_row(i) == 0) { j = rand() % k; e = matrix_insert_entry(i,j); } if(degree_of_row(i) == 1) { do { j = rand()% k; } while (matrix_has_entry(i,j)); matrix_insert_entry(i,j); } } Roca, et al. Expires December 30, 2005 [Page 13] Internet-Draft LDPC Forward Error Correction June 2005 The right side (the staircase) is generated with the following algorithm: for(i = 0; i < n-k; i++) { /* for each row */ matrix_insert_entry(i,k+i); if (i > 0) matrix_insert_entry(i,k+i-1); } 6.3 Encoding Thanks to the staircase matrix, repair symbol creation is straightforward: each repair symbol is equal to the sum of all source symbols in the associated equation, plus the previous repair packet. Therefore encoding should follow the natural repair symbol order, i.e. generate repair symbol with ESI i before symbol ESI i+1. 6.4 Decoding Decoding can be done using the general LDPC iterative decoding algorithm as described in Appendix A. Other techniques can be used, for instance solving th system of n-k linear equations whose variables are the source an repair symbols Roca, et al. Expires December 30, 2005 [Page 14] Internet-Draft LDPC Forward Error Correction June 2005 7. Full Specification of the LDPC-Triangle Scheme 7.1 Instance Specific Parameters LDPC-Triangle is identified by th Under-Specified FEC Encoding ID 132 and the the FEC Instance ID 1. LDPC-Triangle is based on a pseudo-random number generator as specified in Section 5.4. Therefore the seed used to initiate the PRNG is an instance-specific FEC Object Transmission Information element, and MUST be transmitted within the FEC OTI, as specified in Section 4.2. 7.2 Parity Check Matrix Creation The matrix creation algorithm for LDPC Triangle is the following. The left side is the same as for LDPC Staircase (see Section 6.2). The right side (the triangle) is generated with the following algorithm: for(i = 0; i < n-k; i++) { /* for each row */ /* create the identity */ matrix_insert_entry(i,k+i); if (i > 0) { /* create the staircase */ matrix_insert_entry(i,k+i-1); /* fill the triangle */ int j = i; for (l = 0; l < j; l++) { if (j != 0) { temp = rand() % j; matrix_insert_entry(pchkMatrix, i, k+j); } } } } 7.3 Encoding Just like LDPC-Triangle repair symbol creation is straightforward: each repair symbol is equal to the sum of all source symbols in the associated equation, plus some previous repair packets specified in the triangle. Encoding should follow the natural repair symbol order, i.e. generate repair symbol with ESI i before symbol ESI i+1. Roca, et al. Expires December 30, 2005 [Page 15] Internet-Draft LDPC Forward Error Correction June 2005 7.4 Decoding Decoding can be done using the general LDPC iterative decoding algorithm as described in Appendix A. Other techniques can be used, for instance solving th system of n-k linear equation whose variables are the source an repair symbols Roca, et al. Expires December 30, 2005 [Page 16] Internet-Draft LDPC Forward Error Correction June 2005 8. Security Considerations The security considerations for this document are the same as they are for RFC 3452 [RFC3452]. Roca, et al. Expires December 30, 2005 [Page 17] Internet-Draft LDPC Forward Error Correction June 2005 9. Intellectual Property The authors are not aware of any intellectual property rights associated to the two LDPC codes specified within this document. Yet other LDPC codes and associated techniques MAY be covered by IPR. Roca, et al. Expires December 30, 2005 [Page 18] Internet-Draft LDPC Forward Error Correction June 2005 10. Acknowledgments Section 5.3 is derived from a previous Internet-Draft, and we would like to thank S. Peltotalo and J. Peltotalo for their contribution. We would also like to thank Pascal Moniot from STMicroelectronics for his comments. Roca, et al. Expires December 30, 2005 [Page 19] Internet-Draft LDPC Forward Error Correction June 2005 11. References 11.1 Normative References [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119, BCP 14, March 1997. [RFC3452] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M., and J. Crowcroft, "Forward Error Correction (FEC) Building Block", RFC 3452, December 2002. [RFC3453] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M., and J. Crowcroft, "The Use of Forward Error Correction (FEC) in Reliable Multicast", RFC 3453, December 2002. [fec-bb-revised] Watson, M., Luby, M., and L. Vicisano, "Forward Error Correction (FEC) Building Block (revised)", draft-ietf- rmt-fec-bb-revised-00.txt draft-ietf-rmt-fec-bb-revised- 00.txt, April 2005. 11.2 Informative References [LDPCrefimpl] Roca, V., Neumann, C., and J. Laboure, "LDPC-Staircase/ LDPC-Triangle Codec Reference Implementation", MCLv3 project PLANETE Research Team, INRIA Rhone-Alpes, June 2005. [Mac03] MacKay, D., "Information Theory, Inference and Learning Algorithms", Cambridge University Press, ISBN: 0521642981, 2003. [Roca04] Roca, V. and C. Neumann, "Design, Evaluation and Comparison of Four Large Block FEC Codecs: LDPC, LDGM, LDGM Staircase and LDGM Triangle, Plus a Reed-Solomon Small Block FEC Codec", INRIA Research Report RR-5225, June 2004. Roca, et al. Expires December 30, 2005 [Page 20] Internet-Draft LDPC Forward Error Correction June 2005 Authors' Addresses Vincent Roca INRIA 655, av. de l'Europe Zirst; Montbonnot ST ISMIER cedex 38334 France Phone: Email: vincent.roca@inrialpes.fr URI: Christoph Neumann INRIA 655, av. de l'Europe Zirst; Montbonnot ST ISMIER cedex 38334 France Phone: Email: christoph.neumann@inrialpes.fr URI: David Furodet STMicroelectronics 12, Rue Jules Horowitz BP217 Grenoble Cedex 38019 France Phone: Email: david.furodet@st.com URI: Roca, et al. Expires December 30, 2005 [Page 21] Internet-Draft LDPC Forward Error Correction June 2005 Appendix A. Iterative Decoding Algorithm (Informative) LDPC decoding over a packet erasure channel can be achieved through a trivial iterative decoding algorithm. The underlying idea is the following: Given a set of linear equations, if one of them has only one remaining unknown variable, then the value of this variable is that of the constant term. So, replace this variable by its value in all remaining linear equations, and reiterate. The value of several variables can therefore be found by this recursive algorithm. Applied to LDPC FEC codes working over an erasure packet, the parity check matrix defines a set of linear equations. The variables are the source symbols and repair symbols. Of course, from a decoding point of view, finding (i.e. decoding) all source symbols is the target. Finding repair symbols is often required to that purpose, but this is not the final goal. The iterative decoding algorithm is the following: Initialization: allocate a partial sum buffer partial_sum_i for each line i: set it to 0. For each newly received or decoded symbol s_i with ESI i: 1. If s_i is an already decoded or received symbol, return immediately and do nothing. 2. If s_i is a source symbol, it is permanently stored in memory. 3. For each equation j having a degree greater than one (i.e. more than one unknown variable), with an entry in column i (i.e. having s_i as a variable), do the following: + add s_i to partial_sum_i; + remove the entry (j, i) of the H matrix. + If the new degree of equation j is one, we have decoded a new packet and have to remember the index of the equation in a list of indexes for newly decoded packets for step 4. 4. For all newly generated packets in step 3: + remove the last entry in equation j, Roca, et al. Expires December 30, 2005 [Page 22] Internet-Draft LDPC Forward Error Correction June 2005 + move partial_sum_j into th buffer of symbol s_l, + goto step 1 with the newly created symbol s_l Roca, et al. Expires December 30, 2005 [Page 23] Internet-Draft LDPC Forward Error Correction June 2005 Intellectual Property Statement The IETF takes no position regarding the validity or scope of any Intellectual Property Rights or other rights that might be claimed to pertain to the implementation or use of the technology described in this document or the extent to which any license under such rights might or might not be available; nor does it represent that it has made any independent effort to identify any such rights. Information on the procedures with respect to rights in RFC documents can be found in BCP 78 and BCP 79. Copies of IPR disclosures made to the IETF Secretariat and any assurances of licenses to be made available, or the result of an attempt made to obtain a general license or permission for the use of such proprietary rights by implementers or users of this specification can be obtained from the IETF on-line IPR repository at http://www.ietf.org/ipr. The IETF invites any interested party to bring to its attention any copyrights, patents or patent applications, or other proprietary rights that may cover technology that may be required to implement this standard. Please address the information to the IETF at ietf-ipr@ietf.org. Disclaimer of Validity This document and the information contained herein are provided on an "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Copyright Statement Copyright (C) The Internet Society (2005). This document is subject to the rights, licenses and restrictions contained in BCP 78, and except as set forth therein, the authors retain all their rights. Acknowledgment Funding for the RFC Editor function is currently provided by the Internet Society. Roca, et al. Expires December 30, 2005 [Page 24]