Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes ††thanks: Presented at The 34th Symposium on Information Theory and Its Applications (SITA2011), pp.405–410, Ousyuku, Iwate, Japan, Nov. 29–Dec. 2, 2011.
In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed–Solomon codes and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Gröbner basis, are canonically isomorphic, and moreover, the isomorphism is given by the extension through linear feedback shift registers from Gröbner basis and discrete Fourier transforms. Next, the lemma is applied to fast unified system of encoding and decoding erasures and errors in a certain class of affine variety codes.
Keywords: Berlekamp–Massey–Sakata algorithm, Gröbner basis, discrete Fourier transforms, order domain codes, evaluation codes.
Despite many researches have been done for both of encoding and erasure-and-error decoding, any relation between them have never been found so far except maximum distance separable (MDS) codes, which satisfy , where , , and is the code length, dimension, and the minimum distance of the code, respectively. Since the correctable numbers of erasures and of errors satisfy , it is obvious for MDS codes that, if , then erasure-only decoding can determine redundant symbols, that is, systematic encoding is done by erasure-only decoding. In general, algebraic geometry codes are not MDS codes; there has never been known no such case where erasure decoding can undertake systematic encoding.
In this paper, we first establish a lemma that is essential in algebraic coding theory. We observe that almost all manipulations on algebraic Goppa codes such as encoding and decoding are described in terms of the lemma, which provides an isomorphism between a certain pair of vector spaces over a finite field. The isomorphism of the lemma is written by the combined map of the extension by the linear recurrence relation from a Gröbner basis and -dimensional inverse discrete Fourier transform (IDFT). Next, the lemma is applied to a class of affine variety codes ,, which are essentially same as order domain codes or evaluation codes ,,, and enables us to decode efficiently erasures and errors. Finally, we notice that, in a class of affine variety codes, systematic encoding can be viewed as a certain type of erasure-only decoding.
The rest of this paper is organized as follows. In Section II, we prepare notations. In Section III, we state the lemma. In Subsection III-A, we generalize discrete Fourier transforms (DFTs) from on into on . In Subsection III-B, two vector spaces are defined via Gröbner basis. In Subsection III-C, we give an isomorphism between the vector spaces. In Section IV, we apply the lemma. In Subsection IV-A, we construct affine variety codes in terms of the lemma. In Subsection IV-B, an erasure-and-error decoding algorithm and its relation with systematic encoding is described. In Section V, the number of finite-field operations in our algorithm is estimated. Section VI concludes the paper.
Throughout this paper, is the set of non-negative integers and is a fixed primitive element of finite field , where is a prime power. For with , let . For two sets and , is defined as . For arbitrary finite set , let denote a vector space over whose componensts are indexed by . For any arbitrary subset , the vector space is considered as a subspace of by .
Iii Main lemma
Iii-a Fourier-type transforms on
Let be a positive integer and let
In this subsection, Fourier-type transforms are defined as maps between two vector spaces, which are isomorphic to ,
For , discrete Fourier transform (DFT) is defined as 111We consider that means “the index of is ”, i.e., and not .
where is defined by , and is considered as the substituted value , that is, 1 for all if .
The simplest case is . Note that, if and , then trivially holds. Thus can be directly written as
Assume the next simplest case . Then, for each , can be directly written as
Assume . Then, for each , can be directly written as
In general, to write directly, equalities are required.
On the other hand, for , inverse discrete Fourier transform (IDFT) is defined as follows. 222Similarly to footnote 1, means that and not that . For each , a subset of is determined such that and for all . Then, define
where and, for ,
For example, if for , then is equal to , there is only one choice of , and in this case the definition (2) implies
in other words, agrees with -dimensional inverse discrete Fourier transform if is restricted to . In general, for each , is equal to a linear combination of inverse discrete Fourier transforms whose dimensions do not exceed .
Assume . If , then , , and . If , then , , and , respectively. Thus can be directly written as
Assume . For , for example, if , then , , and ; if and , then , , and , repsectively. Thus can be directly written as (4).
Assume . For , for example, if , then . Then has four choices, i.e., , and respectively, . Thus can be directly written as (5). In general, the summand in each condition of consists of terms, where is the number of nonzero components in .
The two linear maps
are inverse each other, that is, and .
Iii-B Two vector spaces and
Let and . 333For any finite set , the number of elements in is represented by . One of the two vector spaces in the lemma is given by
namely, is the vector space over indexed by the elements of , whose dimension is trivially . The other of the two vector spaces is somewhat complicated to define, since it requires Gröbner basis theory. Let be the ring of polynomials with coefficients in whose variables are . Let be an ideal of defined by
We fix a monomial order of . We denote, for ,
whrere for , and is called the leading monomial of . Then the support of for is defined by
where . Fortunately, has an intuitive description if a Gröbner basis of is obtained; it corresponds to the area surrounded by . The support of for is equivalently defined by
Then the other of the two vector spaces is given by
namely, is the vector space over indexed by the elements of . Since is a basis of that is the quotient ring viewed as a vector space over , is isomorphic to . It is known , that the evaluation map
is an isomorphism between two vector spaces. Thus the map (7) is also written as
Since and have the same dimension , it is trivial that is isomorphic to as a vector space over . However, this type of isomorphic maps depends on the choices of the bases of vector spaces; in addition, the normal orthogonal basis is not always convienient for encoding and decoding. Our lemma asserts that there is a canonical isomorphism that does not depend on the bases. As explained in Introduction, the isomorphic map of the lemma is the composition map of the extension and IDFT, which are defined accurately in the next subsection III-C. On the other hand, the inverse map can be written concisely; that is “DFT”
which is actually the compound of DFTs in various dimensions. It is shown from the definitions that the matrices that represent two maps (8) and (9) are transposed each other if the bases of vector spaces are fixed.
Iii-C Isomorphic map
Let be a Gröbner basis for the ideal with respect to . We assume that consists of elements , where
We define that satisfies the linear recurrence relation from if and only if there exists such that, for all and all ,
where the indices of are viewed within if for . Then we denote that .
Namely, each satisfies equations. Then we also say that is the extension of . In fact, there is one-to-one correspondence between arbitrary vectors and all vectors that satisfy the linear recurrence relation from ; from a given , generate inductively by
Then we obtain that satisfies (13); the resulting values do not depend on the order of the generation because of the minimal property of Gröbner bases. Conversely, from a given that satisfies (13), we obtain a vector by restricting to . Thus all that satisfy the linear recurrence relation from are the extension of by (14). Denote as the extension map , and moreover, denote as the restriction map . The following lemma is frequently used in this paper.
If , it holds, for , that .
Moreover, the composition map in the following commutative diagram
V_A & & \rTo^F_N^-1& & V_Ω
\uTo^E& & & & \dTo_R
V_S & & \rTo^C& & V_Ψ
gives an isomorphism between and . The composition map is written as .
Note that, if we admit that is isomorphic to by , then the first assertion of the lemma “ for ” deduces the isomorphism and since the image of by agrees with . We apply this lemma by putting as the set of rational points, the set of erasure-and-error locations, and the set of redundant locations of codewords.
Iv Applications of main lemma
Iv-a Affine variety codes 
and that is a linearly independent basis of . Since in (12) is the value of the inner product for and in , the dual code of is equal to . Thus the dimension or the number of information symbols of is equal to , in other words, .
Consider a subspace of with that has dimension . It follows from the isomorphic map of the lemma that
which is similar to (15). While the definition (12) of is indirect and not constructive, the equality (16) provides a direct construction and it corresponds to a non-systematic encoding of . Moreover, it is shown in the next subsection that the lemma also gives the systematic encoding for a class of such codes.
It is shown  that and represent all linear codes over respectively. Futhermore, the decoding algorithm  using Gröbner basis up to half the minimum distance is shown for . However, since this type of decoding belongs to the class of NP-complete problems , it is strongly suggested that the algorithm in  does not run in polynomial time.
There is another algorithm  that decode all linear codes by -error locating pair and solving system of linear equations. The algorithm  can correct at least up to half the Feng–Rao minimum distance bound and its computational complexity equals O, where is the code length of the linear code and means that for all and some constant .
It is also shown  that all linear codes over are represented as algebraic geometric (AG) codes from algebraic curves. As for fast decoding, Sakata et al.  showed fast algorithm for decoding up to applicable to AG codes of one-point type, a well-studied subclass of AG codes. Sakata et al.  also showed that the similar algorithm to that in  can decode erasures and errors up to for one-point AG codes. O’Sullivan , generalized BMS algorithm for finding the Gröbner basis of error locator ideal of affine variety codes. However, fast decoding of affine variety codes including finding error values has been an open problem so far.
Iv-B DFT erasure-and-error decoding and systematic encoding
Consider the encoding problem for . From the lemma, non-systematic encoding is obtained as follows. For , let be its extended vector. Then holds since for all . However, since error-correcting codes are usually encoded systematically, it is natural to consider the systematic encoding for , which is a certain type of erasure-only decoding as we observe in this subsection.
Let so that corresponds to the set of redundant positions and corresponds to the set of information positions. Then holds since and the definition (6). From now on, consider the linear codes , i.e.,
We choose such that . Then holds.
Suppose that erasure-and-error has occurred in a received word from the channel. Let be the set of erasure locations and let be the set of error locations; we suppose that is known but and are unknown, that , and that . If with and holds, where is the Feng–Rao minimum distance bound ,, then it is known that the erasure-and-error version , of Berlekamp–Massey–Sakata (BMS) algorithm ,[References, Chapter 10] or multidimensional Berlekamp–Massey (BM) algorithm calculates the Gröbner basis . Since is known, can be calculated by the ordinary error-only version in advance and then can be calculated by the erasure-and-error version from the syndrome and the initial value