Reliable Multicast Transport J. Lacan Internet-Draft ENSICA/LAAS-CNRS Intended status: Experimental V. Roca Expires: June 25, 2007 INRIA J. Peltotalo S. Peltotalo Tampere University of Technology December 22, 2006 Reed-Solomon Forward Error Correction (FEC) draft-ietf-rmt-bb-fec-rs-02.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 June 25, 2007. Copyright Notice Copyright (C) The IETF Trust (2006). Lacan, et al. Expires June 25, 2007 [Page 1]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 Abstract This document describes a Fully-Specified FEC scheme for the Reed- Solomon forward error correction code and its application to the reliable delivery of data objects on the packet erasure channel. Reed-Solomon codes belong to the class of Maximum Distance Separable (MDS) codes, i.e. they enable a receiver to recover the k source symbols from any set of k received symbols. The implementation described here is compatible with the implementation from Luigi Rizzo. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Definitions Notations and Abbreviations . . . . . . . . . . . 6 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6 3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 7 4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 8 4.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 8 4.2. FEC Object Transmission Information . . . . . . . . . . . 9 4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 9 4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9 4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 9 4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10 5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.1. Determining the Maximum Source Block Length (B) . . . . . 12 5.2. Determining the Number of Encoding Symbols of a Block . . 12 6. Reed-Solomon Codes Specification for the Erasure Channel . . . 14 6.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 14 6.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 15 6.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 15 6.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 16 6.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 16 6.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 16 6.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 17 6.4. Implementation for the Packet Erasure Channel . . . . . . 17 7. Security Considerations . . . . . . . . . . . . . . . . . . . 19 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 20 9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 21 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 22 10.1. Normative References . . . . . . . . . . . . . . . . . . . 22 10.2. Informative References . . . . . . . . . . . . . . . . . . 22 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 24 Lacan, et al. Expires June 25, 2007 [Page 2]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 Intellectual Property and Copyright Statements . . . . . . . . . . 25 Lacan, et al. Expires June 25, 2007 [Page 3]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 1. Introduction The use of Forward Error Correction (FEC) codes is a classical solution to improve the reliability of multicast and broadcast transmissions. The [2] document describes a general framework to use FEC in Content Delivery Protocols (CDP). The companion document [3] describes some applications of FEC codes for content delivery. Recent FEC schemes like [6] and [7] proposed erasure codes based on sparse graphs/matrices. These codes are efficient in terms of processing but not optimal in terms of correction capabilities when dealing with "small" objects. The FEC scheme described in this document belongs to the class of Maximum Distance Separable codes that are optimal in terms of erasure correction capability. In others words, it enables a receiver to recover the k source symbols from any set of exactly k encoding symbols. Even if the encoding/decoding complexity is larger than that of [6] or [7], this family of codes is very useful. Many applications dealing with content transmission or content storage already rely on packet-based Reed-Solomon codes. In particular, many of them use the Reed-Solomon codec of Luigi Rizzo [4]. The goal of the present document to specify an implementation of Reed-Solomon codes that is compatible with this codec. Lacan, et al. Expires June 25, 2007 [Page 4]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 2. Terminology 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 [1]. Lacan, et al. Expires June 25, 2007 [Page 5]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 3. Definitions Notations and Abbreviations 3.1. Definitions This document uses the same terms and definitions as those specified in [2]. Additionally, it uses the following definitions: Source symbol: unit of data used during the encoding process. Encoding symbol: unit of data generated by the encoding process. Repair symbol: encoding symbol that is not a source symbol. Systematic code: FEC code in which the source symbols are part of the encoding symbols. Source block: a block of k source symbols that are considered together for the encoding. Encoding Symbol Group: a group of encoding symbols that are sent together within the same packet, and whose relationships to the source block 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_r denotes the number of repair symbols generated for a source block. n denotes the encoding block length, i.e. the number of encoding symbols generated for a source block. Therefore: n = k + n_r. max_n denotes the maximum number of encoding symbols generated for any source block. B denotes the maximum source block length in symbols, i.e. the maximum number of source symbols per source block. Lacan, et al. Expires June 25, 2007 [Page 6]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 N denotes the number of source blocks into which the object shall be partitioned. E denotes the encoding symbol length in bytes. S denotes the symbol size in units of m-bit elements. When m = 8, then S and E are equal. m defines the length of the elements in the finite field, in bits. q defines the number of elements in the finite field. We have: q = 2^^m in this specification. G denotes the number of encoding symbols per group, i.e. the number of symbols sent in the same packet. GM denotes the Generator Matrix of a Reed-Solomon code. rate denotes the "code rate", i.e. the k/n ratio. a^^b denotes a raised to the power b. a^^-1 denotes the inverse of a. I_k denotes the k*k identity matrix. 3.3. Abbreviations This document uses the following abbreviations: ESI stands for Encoding Symbol ID. FEC OTI stands for FEC Object Transmission Information. RS stands for Reed-Solomon. MDS stands for Maximum Distance Separable code. GF(q) denotes a finite field (A.K.A. Galois Field) with q elements. We assume that q = 2^^m in this document. Lacan, et al. Expires June 25, 2007 [Page 7]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 4. Formats and Codes 4.1. FEC Payload ID The FEC Payload ID is composed of the Source Block Number and the Encoding Symbol ID. The length of these two fields depends on the parameter m (which is transmitted in the FEC OTI) as follows : o The Source Block Number (field of size 32-m bits) identifies from which source block of the object the encoding symbol(s) in the payload is (are) generated. There are a maximum of 2^^(32-m) blocks per object. o The Encoding Symbol ID (field of size m bits) identifies which specific encoding symbol(s) generated from the source block is(are) carried in the packet payload. There are a maximum of 2^^m encoding symbols per block. The first k values (0 to k - 1) identify source symbols, the remaining n-k values identify repair symbols. There MUST be exactly one FEC Payload ID per source or repair packet. In case of an Encoding Symbol Group, 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 from the ESI of the first symbol by incrementing sequentially the ESI. The format of the FEC Payload ID for m = 8 and m = 16 is illustrated in Figure 1 and Figure 2 respectively. 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 (32-8=24 bits) | Enc. Symb. ID | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 1: FEC Payload ID encoding format for m = 8 (default) 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 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Src Block Nb (32-16=16 bits) | Enc. Symbol ID (m=16 bits) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 2: FEC Payload ID encoding format for m = 16 Lacan, et al. Expires June 25, 2007 [Page 8]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 4.2. FEC Object Transmission Information 4.2.1. Mandatory Elements o FEC Encoding ID: the Fully-Specified FEC Scheme described in this document uses FEC Encoding ID 2. 4.2.2. Common Elements The following elements MUST be defined with the present FEC scheme: o Transfer-Length (L): a non-negative integer indicating the length of the object in bytes. There are some restrictions on the maximum Transfer-Length that can be supported : max_transfer_length = 2^^(32-m) * B * E For instance, for m = 8, for B = 2^^8 - 1 (because the codec operates on a finite field with 2^^8 elements) and if E = 1024 bytes, then the maximum transfer length is approximately equal to 2^^42 bytes (i.e. 4 Tera Bytes). Similarly, for m = 16, for B = 2^^16 - 1 and if E = 1024 bytes, then the maximum transfer length is also approximately equal to 2^^42 bytes. For larger objects, another FEC scheme, with a larger Source Block Number field in the FEC Payload ID, could be defined. Another solution consists in fragmenting large objects into smaller objects, each of them complying with the above limits. o Encoding-Symbol-Length (E): a non-negative integer indicating the length of each encoding symbol in bytes. o Maximum-Source-Block-Length (B): a non-negative integer indicating the maximum number of source symbols in a source block. o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer indicating the maximum number of encoding symbols generated for any source block. Section 5 explains how to derive the values of each of these elements. 4.2.3. Scheme-Specific Elements The following element MUST be defined with the present FEC Scheme. It contains two distinct pieces of information: Lacan, et al. Expires June 25, 2007 [Page 9]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 o G: a non-negative integer indicating the number of encoding symbols per group used for the object. The default value is 1, meaning that each packet contains exactly one symbol. When no G parameter is communicated to the decoder, then this latter MUST assume that G = 1. o Finite Field parameter, m: The m parameter is the length of the finite field elements, in bits. It also characterizes the number of elements in the finite field: q = 2^^m elements. The default value is m = 8. When no finite field size parameter is communicated to the decoder, then this latter MUST assume that m = 8. 4.2.4. Encoding Format This section shows two possible encoding formats of the above FEC OTI. The present document does not specify when one encoding format or the other should be used. 4.2.4.1. Using the General EXT_FTI Format The FEC OTI binary format is the following, when the EXT_FTI mechanism is used (e.g. within the ALC [8] or NORM [9] protocols). 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) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | m | G | Encoding Symbol Length (E) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Max Source Block Length (B) | Max Nb Enc. Symbols (max_n) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 3: EXT_FTI Header Format 4.2.4.2. Using the FDT Instance (FLUTE specific) When it is desired that the FEC OTI be carried in the FDT Instance of a FLUTE session [10], the following XML attributes must be described for the associated object: o FEC-OTI-FEC-Encoding-ID o FEC-OTI-Transfer-Length (L) Lacan, et al. Expires June 25, 2007 [Page 10]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 o FEC-OTI-Encoding-Symbol-Length (E) o FEC-OTI-Maximum-Source-Block-Length (B) o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n) o FEC-OTI-Scheme-Specific-Info The FEC-OTI-Scheme-Specific-Info contains the string resulting from the Base64 encoding (in the XML Schema xs:base64Binary sense) of the following value: 0 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | m | G | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 4: FEC OTI Scheme Specific Information to be Included in the FDT Instance When no m parameter is to be carried in the FEC OTI, the m field is set to 0 (which is not a valid seed value). Otherwise the m field contains a valid value as explained in Section 4.2.3. Similarly, when no G parameter is to be carried in the FEC OTI, the G field is set to 0 (which is not a valid seed value). Otherwise the G field contains a valid value as explained in Section 4.2.3. When neither m nor G are to be carried in the FEC OTI, then the sender simply omits the FEC-OTI-Scheme-Specific-Info attribute. After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific Information are transformed into a string of 4 printable characters (in the 64-character alphabet) and added to the FEC-OTI-Scheme- Specific-Info attribute. Lacan, et al. Expires June 25, 2007 [Page 11]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 5. Procedures 5.1. Determining the Maximum Source Block Length (B) The finite field size parameter, m, defines the number of non zero elements in this field which is equal to: q - 1 = 2^^m - 1. Note that q - 1 is also the theoretical maximum number of encoding symbols that can be produced for a source block. For instance, when m = 8 (default): max1_B = 2^^8 - 1 = 255 Additionally, a codec MAY impose other limitations on the maximum block size. Yet it is not expected that such limits exist when using the default m = 8 value. This decision MUST be clarified at implementation time, when the target use case is known. This results in a max2_B limitation. Then, B is given by: B = min(max1_B, max2_B) Note that this calculation is only required at the coder, since the B parameter is communicated to the decoder through the FEC OTI. 5.2. Determining the Number of Encoding Symbols of a Block The following algorithm, also called "n-algorithm", explains how to determine the actual number of encoding symbols for a given block. AT A SENDER: Input: B: Maximum source block length, for any source block. Section 5.1 explains how to determine its value. k: Current source block length. This parameter is given by the block partitioning algorithm. rate: FEC code rate, which is given by the user (e.g. when starting a FLUTE sending application). It is expressed as a floating point value. Output: max_n: Maximum number of encoding symbols generated for any source block Lacan, et al. Expires June 25, 2007 [Page 12]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 n: Number of encoding symbols generated for this source block Algorithm: max_n = floor(B / rate); if (max_n > 2^^m - 1) then return an error ("invalid code rate"); n = floor(k * max_n / B); AT A RECEIVER: Input: B: Extracted from the received FEC OTI max_n: Extracted from the received FEC OTI k: Given by the block partitioning algorithm Output: n Algorithm: n = floor(k * max_n / B); Note that a Reed-Solomon decoder does not need to know the n value. Therefore the receiver part of the "n-algorithm" is not necessary from the Reed-Solomon decoder point of view. Yet a receiving application using the Reed-Solomon FEC scheme will sometimes need to know the n value used by the sender, for instance for memory management optimizations. To that purpose, the FEC OTI carries all the parameters needed for a receiver to execute the above algorithm. Lacan, et al. Expires June 25, 2007 [Page 13]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 6. Reed-Solomon Codes Specification for the Erasure Channel Reed-Solomon (RS) codes are linear block codes. They also belong to the class of MDS codes. A [n,k]-RS code encodes a sequence of k source elements defined over a finite field GF(q) into a sequence of n encoding elements, where n is upper bounded by q - 1. The implementation described in this document is based on a generator matrix built from a Vandermonde matrix put into systematic form. Section 6.1 to Section 6.3 specify the [n,k]-RS codes when applied to m-bit elements, and Section 6.4 the use of [n,k]-RS codes when applied to symbols composed of several m-bit elements, which is the target of this specification. 6.1. Finite Field A finite field GF(q) is defined as a finite set of q elements which has a structure of field. It contains necessarily q = p^^m elements, where p is a prime number. With packet erasure channels, p is always set to 2. The elements of the field GF(2^^m) can be represented by polynomials with binary coefficients (i.e. over GF(2)) of degree lower or equal than m-1. The polynomials can be associated to binary vectors of length m. For example, the vector (11001) represents the polynomial 1 + x + x^^4. This representation is often called polynomial representation. The addition between two elements is defined as the addition of binary polynomials in GF(2) and the multiplication is the multiplication modulo a given irreducible polynomial over GF(2) of degree m with coefficients in GF(2). Note that all the roots of this polynomial are in GF(2^^m) but not in GF(2). A finite field GF(2^^m) is completely characterized by the irreducible polynomial. The following polynomials are chosen to represent the field GF(2^^m), for m varying from 2 to 16: m = 2, "111" (1+x+x^^2) m = 3, "1101", (1+x+x^^3) m = 4, "11001", (1+x+x^^4) m = 5, "101001", (1+x^^2+x^^5) m = 6, "1100001", (1+x+x^^6) m = 7, "10010001", (1+x^^3+x^^7) Lacan, et al. Expires June 25, 2007 [Page 14]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8) m = 9, "1000100001", (1+x^^4+x^^9) m = 10, "10010000001", (1+x^^3+x^^10) m = 11, "101000000001", (1+x^^2+x^^11) m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12) m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13) m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14) m = 15, "1100000000000001", (1+x+x^^15) m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16) In order to facilitate the implementation, these polynomials are also primitive. This means that any element of GF(2^^m) can be expressed as a power of a given root of this polynomial. These polynomials are also chosen so that they contain the minimum number of monomials. 6.2. Reed-Solomon Encoding Algorithm 6.2.1. Encoding Principles Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over GF(2^^m). Let e = (e_0, ..., e_{n-1}) be the corresponding encoding vector of n elements over GF(2^^m). Being a linear code, encoding is performed by multiplying the source vector by a generator matrix, GM, of k rows and n columns over GF(2^^m). Thus: e = s * GM. The definition of the generator matrix completely characterizes the RS code. Let us consider that: n = 2^^m - 1 and: 0 < k <= n. Let us denote by alpha the root of the primitive polynomial of degree m chosen in the list of Section 6.1 for the corresponding value of m. Let us consider a Vandermonde matrix of k rows and n columns, denoted by V_{k,n}, and built as follows: the {i, j} entry of V_{k,n} is v_{i,j} = alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1. This matrix generates a MDS code. However, this MDS code is not systematic, which is a problem for many networking applications. To obtain a systematic matrix (and code), the simplest solution consists in considering the matrix V_{k,k} formed by the first k columns of Lacan, et al. Expires June 25, 2007 [Page 15]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 V_{k,n}, then to invert it and to multiply this inverse by V_{k,n}. Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity matrix I_k on its first k columns, meaning that the first k encoding elements are equal to source elements. Besides the associated code keeps the MDS property. Therefore, the generator matrix of the code considered in this document is: GM = (V_{k,k}^^-1) * V_{k,n} Note that, in practice, the [n,k]-RS code can be shortened to a [n',k]-RS code, where k <= n' < n, by considering the sub-matrix formed by the n' first columns of GM. 6.2.2. Encoding Complexity Encoding can be performed by first pre-computing GM and by multiplying the source vector (k elements) by GM (k rows and n columns). The complexity of the pre-computation of the generator matrix can be estimated as the complexity of the multiplication of the inverse of a Vandermonde matrix by n-k vectors (i.e. the last n-k columns of V_{k,n}). Since the complexity of the inverse of a k*k- Vandermonde matrix by a vector is O(k * log^^2(k)), the generator matrix can be computed in 0((n-k)* k * log^^2(k)) operations. When the genarator matrix is pre-computed, the encoding needs k operations per repair element (vector-matrix multiplication). Encoding can also be performed by first computing the product s * V_{k,k}^^-1 and then by multiplying the result with V_{k,n}. The multiplication by the inverse of a square Vandermonde matrix is known as the interpolation problem and its complexity is O(k * log^^2 (k)). The multiplication by a Vandermonde matrix, known as the multipoint evaluation problem, requires O((n-k) * log(k)) by using Fast Fourier Transform, as explained in [11]. The total complexity of this encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k)) operations per repair element. 6.3. Reed-Solomon Decoding Algorithm 6.3.1. Decoding Principles The Reed-Solomon decoding algorithm for the erasure channel allows the recovery of the k source elements from any set of k received elements. It is based on the fundamental property of the generator matrix which is such that any k*k-submatrix is invertible (see [5]). The first step of the decoding consists in extracting the k*k submatrix of the generator matrix obtained by considering the columns Lacan, et al. Expires June 25, 2007 [Page 16]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 corresponding to the received elements. Indeed, since any encoding element is obtained by multiplying the source vector by one column of the generator matrix, the received vector of k encoding elements can be considered as the result of the multiplication of the source vector by a k*k submatrix of the generator matrix. Since this submatrix is invertible, the second step of the algorithm is to invert this matrix and to multiply the received vector by the obtained matrix to recover the source vector. 6.3.2. Decoding Complexity The decoding algorithm described previously includes the matrix inversion and the vector-matrix multiplication. With the classical Gauss-Jordan algorithm, the matrix inversion requires O(k^^3) operations and the vector-matrix multiplication is performed in O(k^^2) operations. This complexity can be improved by considering that the received submatrix of GM is the product between the inverse of a Vandermonde matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V' which is a submatrix of V_(k,n)). The decoding can be done by multiplying the received vector by V'^^-1 (interpolation problem with complexity O( k * log^^2(k)) ) then by V_{k,k} (multipoint evaluation with complexity O(k * log(k))). The global decoding complexity is then O(log^^2(k)) operations per source element. 6.4. Implementation for the Packet Erasure Channel In a packet erasure channel, each packet (and symbol(s) since packets contain G >= 1 symbols) is either correctly received or erased. The location of the erased symbols in the sequence of symbols must be known. The following specification describes the use of Reed-Solomon codes for generating redundant symbols from the k source symbols and to recover the source symbols from any set of k received symbols. The k source symbols of a source block are assumed to be composed of S m-bit elements. Each m-bit element is associated to an element of the finite field GF(2^^m) through the polynomial representation (Section 6.1). If some of the source symbols contain less than S elements, they are virtually padded with zero elements (it can be the case for the last symbol of the last block of the object). The encoding process produces n-k repair symbols of size S m-bit elements, the k source symbols being also part of the n encoding symbols (Figure 5). These repair symbols are created m-bit element per m-bit element. More specifically, the j-th source vector is composed of the j-th element of each of the source symbols. Similarly, the j-th encoding vector is composed of the j-th element Lacan, et al. Expires June 25, 2007 [Page 17]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 of each encoding symbol. n +-+-+----+-+ +---------------+ +-+-+-----------+-+ 0 | | | | | | | | | | | | | | | | | * | generator | = | | | | | | | | | | | matrix | | | | | | | | | | | | GM | | | | | | source +--------------+ | (k x n) | +---------------------+ vector | | | | | | | +---------------+ ->| | | | | | | j +--------------+ / +---------------------+ | | | | | / | | | | | | | | | | encoding | | | | | | | | | | vector | | | | | | | | | | j | | | | | | | | | | | | | | | S-1 | | | | | | | | | | +-+-+----+-+ +-+-+-----------+-+ k source symbols n encoding symbols (source + repair) Figure 5: Packet encoding scheme An asset of this scheme is that the loss of some encoding symbols produces the same erasure pattern for each of the S encoding vectors. It follows that the matrix inversion must be done only once and will be used by all the S encoding vectors. For large S, this matrix inversion cost becomes negligible in front of the S matrix-vector multiplications. Another asset is that the n-k repair symbols can be produced on demand. For instance, a sender can start by producing a limited number of repair symbols and later on, depending on the observed erasures on the channel, decide to produce additional repair symbols, up to the n-k upper limit. Indeed, to produce the repair symbol e_j, where k <= j < n, it is sufficient to multiply the S source vectors with column j of GM. Lacan, et al. Expires June 25, 2007 [Page 18]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 7. Security Considerations The security considerations for this document are the same as that of [2]. Lacan, et al. Expires June 25, 2007 [Page 19]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 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 [2]. This document assigns the Fully- Specified FEC Encoding ID 2 under the ietf:rmt:fec:encoding name- space to "Reed-Solomon Codes". Lacan, et al. Expires June 25, 2007 [Page 20]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 9. Acknowledgments The authors want to thank Luigi Rizzo for comments on the subject and for the design of the reference Reed-Solomon codec. Lacan, et al. Expires June 25, 2007 [Page 21]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 10. References 10.1. Normative References [1] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119. [2] Watson, M., Luby, M., and L. Vicisano, "Forward Error Correction (FEC) Building Block", draft-ietf-rmt-fec-bb-revised-04.txt (work in progress), September 2006. [3] 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. 10.2. Informative References [4] Rizzo, L., "Reed-Solomon FEC codec (revised version of July 2nd, 1998), available at http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz", July 1998. [5] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting Codes", North Holland, 1977 . [6] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer, "Raptor Forward Error Correction Scheme", Internet Draft draft-ietf-rmt-bb-fec-raptor-object-04 (work in progress), June 2006. [7] Roca, V., Neumann, C., and D. Furodet, "Low Density Parity Check (LDPC) Forward Error Correction", draft-ietf-rmt-bb-fec-ldpc-04.txt (work in progress), December 2006. [8] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered Coding (ALC) Protocol Instantiation", draft-ietf-rmt-pi-alc-revised-03.txt (work in progress), April 2006. [9] Adamson, B., Bormann, C., Handley, M., and J. Macker, "Negative-acknowledgment (NACK)-Oriented Reliable Multicast (NORM) Protocol", draft-ietf-rmt-pi-norm-revised-03.txt (work in progress), September 2006. [10] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, "FLUTE - File Delivery over Unidirectional Transport", Lacan, et al. Expires June 25, 2007 [Page 22]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 draft-ietf-rmt-flute-revised-02.txt (work in progress), August 2006. [11] Gohberg, I. and V. Olshevsky, "Fast algorithms with preprocessing for matrix-vector multiplication problems", Journal of Complexity, pp. 411-427, vol. 10, 1994 . Lacan, et al. Expires June 25, 2007 [Page 23]

Internet-Draft Reed-Solomon Forward Error Correction December 2006 Authors' Addresses Jerome Lacan ENSICA/LAAS-CNRS 1, place Emile Blouin Toulouse 31056 France Email: jerome.lacan@ensica.fr URI: http://dmi.ensica.fr/auteur.php3?id_auteur=5 Vincent Roca INRIA 655, av. de l'Europe Zirst; Montbonnot ST ISMIER cedex 38334 France Email: vincent.roca@inrialpes.fr URI: http://planete.inrialpes.fr/~roca/ Jani Peltotalo Tampere University of Technology P.O. Box 553 (Korkeakoulunkatu 1) Tampere FIN-33101 Finland Email: jani.peltotalo@tut.fi URI: http://atm.tut.fi/mad Sami Peltotalo Tampere University of Technology P.O. Box 553 (Korkeakoulunkatu 1) Tampere FIN-33101 Finland Email: sami.peltotalo@tut.fi URI: http://atm.tut.fi/mad Lacan, et al. Expires June 25, 2007 [Page 24]

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Internet-Draft Reed-Solomon Forward Error Correction December 2006
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