Generalized Hamming weights for linear codes

Motivated by cryptographical applications, the algebraic structure, of linear codes from a new perspective is studied. By viewing the minimum Hamming weight as a certain minimum property of one-dimensional subcodes, a generalized notion of higher-dimensional Hamming weights is obtained. These weights characterize the code performance on the wire-tap channel of type II. Basic properties of generalized weights are derived, the values of these weights for well-known classes of codes are determined, and lower bounds on code parameters are obtained. Several open problems are also listed     

Fast erasure-and-error decoding of algebraic geometry codes up tothe Feng-Rao bound

This article gives an errata (that is erasure- and error-) decoding algorithm of one-point algebraic-geometry codes up to the Feng-Rao (1994) designed minimum distance using Sakata's (see Proc. 1995 IEEE Int. Symp. Information Theory, Whistler, BC, Canada, 1995) multidimensional generalization of the Berlekamp-Massey (1969) algorithm and the voting procedure of Feng and Rao     

关于BCH码的广义Hamming重量上,下限

一个线性码的第ｒ广义Ｈａｍｍｉｎｇ重量是它任意ｒ维子码的最小支集大小。本文给出了一般（本原、狭义）ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量下限和一类ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量上限     

广义Hamming重量上,下界的对偶定理

本文给出了一种广义Ｈａｍｍｉｎｇ重量上、下界的对偶定理。即若给定一个码的对偶码的广义Ｈａｍｍｉｎｇ重量上界（或者下界），可以给出该码的广义Ｈａｍｍｉｎｇ重量上界（或者下界）。Ｈ．Ｓｔｉｃｈ－ｎｏｔｈ（１９９４）曾给出了迹码（如ＢＣＨ码和Ｇｏｐｐａ码的对偶码）的广义Ｈａｍｍｉｎｇ重量一种上、下界，如果采用本文结果就可以给出迹码的对偶码的广义Ｈａｍｍｉｎｇ重量一种上、下界。因此，本文的结果是Ｈ．Ｓｔｉｃｈｎｏｔｈ的结果的有益补充     

Parameters of Goppa codes revisited

We discuss parameters of Goppa (1970) codes, such as minimum distance, covering radius, distance distribution, and generalized Hamming weights. By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds on these parameters     

Comment on “Error rate performance of the projectioncode”

For original paper see ibid., vol.44, p.413-15 (1996 April). The present authors provide counterexamples to the expressions for the minimum Hamming distance of the projection codes in the original paper of Li et al     

ON THE DECODING OF ALGEBRAIC GEOMETRIC CODES BASED ON FIA

Suppose C is an irreducible algebraic curve of genus g, C*(D,G) is an algebraic geometric code with designed minimum distance d* = deg(G)-2g 2. In this paper, a decoding algorithm based on Fundamental Iterative Algorithm(FIA) is presented, also its reasonableness is proved. In fact, our decoding algorithm is a modification of the algorithm proposed by G. L. Fend and T. R. N. Rao(1993) and can correct any received words with errors not more than (d*-1)/2, whereas the complexity is only about one half as much as Feng and Rao's. The procedure can be implemented easily by hardware or software.     

关于Goppa码、BCH码的广义Hamming重量

本文研究了Goppa码、BCH码的广义Hamming重量,给出了Goppa码的广义Hamming重量的一个下界以及求该下界的一个算法;对于本原、狭义BCH码,给出了后面一些广义Hamming重量的确切值。     

二元Goppa码最小距离的新下限

二元Goppa码是一大类很有用的纠错码。但是如何求二元Goppa码的真正最小距离至今没有解决。本文将导出二元Goppa码最小距离的新下限,这个新下限改进了Y.Sugiyama等(1976)和作者(1983)文章的结果。本文的方法不难推广到其他Goppa码中去。     

Approximately Gaussian weight distribution of the iterated productof single-parity-check codes

It has been argued by Battail (see Ann. Telecommun., vol. 44, no. 7/8, p. 392-404, 1989) that for the goodness of long block codes a better criterion to use than the minimum Hamming distance is the closeness of the normalised weight distribution of the code to a Gaussian distribution. Certain iterated-product codes have recently been shown to provide good performance in spite of having a relatively poor Hamming distance. The authors substantiate the arguments of Battail by showing that these iterated-product codes have a weight distribution that is approximately Gaussian even for relatively short lengths     

On the second generalized Hamming weight of the dual code of adouble-error-correcting binary BCH code

The generalized Hamming weight of a linear code is a new notion of higher dimensional Hamming weights. Let C be an [n,k] linear code and D be a subcode. The support of D is the cardinality of the set of not-always-zero bit positions of D. The rth generalized Hamming weight of C, denoted by d_r(C), is defined as the minimum support of an r-dimensional subcode of C. It was shown by Wei (1991) that the generalized Hamming weight hierarchy of a linear code completely characterizes the performance of the code on the type II wire-tap channel defined by Ozarow and Wyner (1984). In the present paper the second generalized Hamming weight of the dual code of a double-error-correcting BCH code is derived and the authors prove that except for m=4, the second generalized Hamming weight of [2^m-1, 2m]-dual BCH codes achieves the Griesmer bound     

Exponential sums and Goppa codes. II

For pt.I, see Proc. AMS, vol.III, p.523-31 (1991). The minimum distance of a Goppa code is found when the length of code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri (1966). This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in other words, the covering radii are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code is found. Finally, a bound on the minimum distance of the dual code is discussed     

On the generalized Hamming weights of several classes of cycliccods

The generalized Hamming weights of a linear code are fundamental code parameters related to the minimal overlap structures of the subcodes. They were introduced by V.K. Wei (1991) and shown to characterize the performance of the linear code in certain cryptographical applications. Results are presented on the generalized Hamming weights of several classes of binary cyclic codes, including primitive double-error-correcting and triple-error-correcting BCH codes, certain reversible cyclic codes, and some extended binary Goppa codes. In particular, the second generalized Hamming weight of primitive double-error-correcting BCH codes is determined and upper and lower bounds are obtained for the generalized Hamming weights for the codes studied. These bounds are compared to results from other methods     

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers.     

On generalized low-density parity-check codes based on Hammingcomponent codes

In this paper we investigate a generalization of Gallager's (1963) low-density (LD) parity-check codes, where as component codes single error correcting Hamming codes are used instead of single error detecting parity-check codes. It is proved that there exist such generalized low-density (GLD) codes for which the minimum distance is growing linearly with the block length, and a lower bound of the minimum distance is given. We also study iterative decoding of GLD codes for the communication over an additive white Gaussian noise channel. The performance in terms of the bit error rate, obtained by computer simulations, is presented for GLD codes of different lengths     

Reply to comments on “Error rate performance of theprojection code”

The authors reply to the comments by Koppelaar and Tolhuizen (see ibid. vol.46, no.9, p.573, 1998). They state that the optimum choice of the slopes used to perform the encoding process in a projection code always results in maximizing the minimum Hamming distance so that d=2 ^r     

LDPC codes from generalized polygons

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.     

Error performance analysis for reliability-based decodingalgorithms

The statistical approach proposed by Agrawal and Vardy (see ibid., vol.46, no.1, p.60-83, 2000) to evaluate the error performance of the generalized minimum distance (GMD) decoding is extended to other reliability-based decoding algorithms for binary linear block codes, namely Chase (1972) type, combined GMD and Chase type, and order statistic decoding (OSD). In all cases, tighter and simpler bounds than those previously proposed have been obtained with this approach     

Combinatorial bounds for list decoding

Informally, an error-correcting code has "nice" list-decodability properties if every Hamming ball of "large" radius has a "small" number of codewords in it. We report linear codes with nontrivial list-decodability: i.e., codes of large rate that are nicely list-decodable, and codes of large distance that are not nicely list-decodable. Specifically, on the positive side, we show that there exist codes of rate R and block length n that have at most c codewords in every Hamming ball of radius H^-1(1-R-1/c)·n. This answers the main open question from the work of Elias (1957). This result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors, improving previous results of Guruswami and Sudan (see IEEE Trans. Inform. Theory, vol.45, p.1757-67, Sept. 1999, and Proc. 32nd ACM Symp. Theory of Computing (STOC), Portland, OR, p. 181-190, May 2000) in this vein. Specifically, for every ε > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate Ω(ε⁴) that can be list-decoded in polynomial time from up to a fraction (1/2-ε) of errors, using lists of size O(ε^-2). On the negative side, we show that for every δ and c, there exists τ < δ, c₁ > 0, and an infinite family of linear codes {C_i}_i such that if n_i denotes the block length of C_i, then C _i has minimum distance at least δ · n_i and contains more than c₁ · n_i^c codewords in some Hamming ball of radius τ · n_i. While this result is still far from known bounds on the list-decodability of linear codes, it is the first to bound the "radius for list-decodability by a polynomial-sized list" away from the minimum distance of the code     

A new lower bound for the minimum distance of binary Goppa codes

Binary Goppa codes are a large and powerful family of error-correcting codes. But how to find the true minimum distance of binary Goppa codes is not solved yet. In this paper a new lower bound for the minimum distance of binary Goppa codes is shown. This new lower bound improves the results in Y. Sugiyama (1976) and Feng Guiliang's (1983) papers. The method in this paper can be generalized to other Goppa codes easily.