True Minimum Distance Research Articles

On Hard and Soft Decision Decoding of BCH Codes

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.

A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity. ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

From ds-Bounds for Cyclic Codes to True Minimum Distance for Abelian Codes

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of ${\mathbb B}$ -apparent distance. We also study conditions for an Abelian code to verify that its ${\mathbb B}$ -apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application.

NONBINARY QUANTUM CODES DERIVED FROM REPEATED-ROOT CYCLIC CODES

Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.

Minimizing Great-Circle Distance Ratios of Undesired and Desired Signal Paths on a Spherical Earth

Preventing interference among mobile radio (MR) nets often requires spectrum managers to observe channel-assignment rules based on minimum ratios of the great-circle distances traversed by desired and undesired (potentially interfering) signals. Finding the true minimum values of these ratios, rather than the simply computed worst-case estimates, is necessary to utilize the spectrum efficiently. This paper provides an analysis of the case where the MRs operate in or above circular service areas on the surface of a spherical Earth. The analysis provides an accurate and efficient way, which is less complicated than performing the calculations on the sphere, to compute true minimum distance ratios. The method uses stereographic projection to transform the original minimization problem into a simpler problem of minimizing a ratio of Euclidean distances, which is expressed as a function of a single real variable, over the boundaries of discs (i.e., circles) in the complex plane. In a large, complicated problem, this methodology has been shown to reduce the computational time by two thirds.

Distance upper bounds and true minimum distance results for turbo-codes designed with DRP interleavers

This paper presents a number of distance upper bounds for Turbo-codes designed with dithered relative prime (drp) interleavers. These bounds help determine how much dither should be applied and which input data patterns to test when designing high-distance interleavers with different lengths. Using a block length of only 640 data bits, true minimum distances of 30, 42 and 53 have been achieved for rate 1/3 Turbo-codes with 4, 8 and 16-state constituent codes, respectively. Simulated error rate results for a block length of 1504 data bits (188 bytes,mpeg packet) show excellent flare performance.

Comparison of shortest sailing distance through random and regular sampling points

Abstract The shortest sailing distance through n sampling points is calculated for simple theoretical sampling domains (square and circle) as well as for a rather irregular and concavely shaped real sampling domain in the Barents Sea. The sampling sites are either located at the nodes of a square grid (regular sampling) or they are randomly distributed. For n less than ten, the exact shortest sailing distance is derived. For larger n, a traveling salesman algorithm (simulated annealing) was applied, and its bias (distance from true minimum) was estimated based on a case where the true minimum distance was known. In general, the average minimum sailing distance based on random sampling was considerably shorter than for regular sampling, and the difference increased with sample size until an asymptotic value was reached at about n=60 for a square domain. For the sampling domain in the Barents Sea used for shrimp (Pandalus borealis) abundance surveys (n=118 stations), the cruise-track lengths based on random sampling were approximately normally distributed. The mean sailing distance was 18% shorter than the cruise track for regular sampling and the standard deviation equalled 2.6%.

On codes from norm–trace curves

The main results of this paper are derived by using only simple Gröbner basis techniques. We present a new construction of evaluation codes from Miura–Kamiya curves C ab . We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometric Goppa codes. In particular we consider codes from norm–trace curves. We show that our estimates give actually the true minimum distance of these codes. The new codes from norm–trace curves perform rather well. In many cases much better than the corresponding geometric Goppa codes. It turns out that an alternative description of the new codes from norm–trace curves can be made by using Høholdt et al.'s in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, Vol. 1, Elsevier, Amsterdam, 1998, pp. 871–961 (Chapter 10) construction of improved dual codes.

Decoding the (47,24,11) quadratic residue code

The techniques needed to decode the (47,24,11) quadratic residue (QR) code differ from the schemes developed for cyclic codes. By finding certain nonlinear relations between the known and unknown syndromes for this special code, two methods are developed to decode up to the true minimum distance of the (47,24,11) QR code. These algorithms can be utilized to decode effectively the 1/2 -rate (48,24,12) QR code for correcting five errors and detecting six errors.

On the minimum distances of Schubert codes

Ghorpade and Lachaud (1998) introduced Schubert codes as codes corresponding to the protective system (Tsfasman and Vladut 1991) defined by Schubert subvarieties of Grassmann varieties Gr(l,m). They gave an upper bound for the minimum distances of these codes in Ghorpade and Lachaud and conjectured their true minimum distances. In this note we prove the conjecture in the case of l=2.

On a sequence of cyclic codes with minimum distance six

A sequence of q-ary cyclic codes is considered. For each finite field GF(q), q/spl ges/4, there is a code with parameters [n, k, d; q]=[q(q-1)+1, q(q-1)-6, 6; q]. We show that all these codes are n-, k-, and d-optimal, with only one exception. Also the dual codes are considered. Their true minimum distances are calculated in the range 4/spl les/q/spl les/32.

On the minimum distance of composite-length BCH codes

We derive a theorem which generalizes Theorem 3 in Chapter 9 of the book "The Theory of Error-Correcting Codes" by F.J. MacWilliams and N.J.A. Sloane (North-Holland, 1977). By this theorem, we are able to give several classes of BCH codes of composite length whose minimum distance does not exceed the BCH bound. Moreover, we show that this theorem can also be used to determine the true minimum distance of some other cyclic codes with composite-length.

Decoding Affine Variety Codes Using Gröbner Bases

We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the F_q-rational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Grobner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.

On the Decoding of Cyclic Codes Using Gröbner Bases

In this paper we revisit an algorithm presented by Chen, Reed, Helleseth, and Troung in [5] for decoding cyclic codes up to their true minimum distance using Grobner basis techniques. We give a geometric characterization of the number of errors, and we analyze the corresponding algebraic characterization. We give a characterization for the error locator polynomial as well. We make these ideas effective using the theory of Grobner bases. We then present an algorithm for computing the reduced Grobner basis over ?2 for the syndrome ideal of cyclic codes, with respect to a lexicographic term ordering. This algorithm does not use Buchberger’s algorithm or the multivariable polynomial division algorithm, but instead uses the form of the generators of the syndrome ideal and an adaptation of the algorithm introduced in [11]. As an application of this algorithm, we present the reduced Grobner basis for the syndrome ideal of the [23, 12, 7] Golay code, and a decoding algorithm.

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the purely lexicographical ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated. >

Studying the locator polynomials of minimum weight codewords of BCH codes

Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2/sup h/-1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2/sup m-2/-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the minimum distance of cyclic codes

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length (with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length \geq 63 .

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.

The minimum distance of all binary cyclic codes of odd lengths from 69 to 99

A computer search has been made to determine the true minimum distance d for all binary cyclic codes having odd lengths n in the range 69\leq n\leq 99 . Using an algorithm originally developed by C. L. Chen, the generator matrix G of each (n,k) binary cyclic code was put in systematic form. All possible codewords obtained from sums of i rows of G , for i= 1,2, \cdots,\upsilon , were examined, and the minimum distance d_{\upsilon} of this set was recorded. Then d=d_{\upsilon} whenever \upsilon >\{(d_{\upsilon}-l)k/n\}-l . Known equivalences among cyclic codes were taken into account, and only one code from each equivalence class was listed. Let g(x) divide x^{n}-1 , where x - 1 is not a factor of g(x) . Then the minimum distances of the codes generated by g(x),(x - l)g(x) and their duals are listed together. For each such set of codes, the value of \upsilon for which a codeword of minimum weight first appeared is listed. The codes found were compared with the list of best codes tabulated by Sloane [5]. Many good cyclic codes have been found. Among the best (n,k,d) cyclic codes found are the following: (73, 27, 20), (73, 36, 16), (85, 12, 34), (85, 20, 28), (87, 31, 22), (89, 56, 11), (91, 51, 14), (93, 20, 32), (93, 23, 29), (93, 31, 24), (93, 33, 22).

Long BCH codes are bad

The BCH bound approaches zero with increasing code length for any fixed value of the ratio of minimum distance to code length. Does this happen because the codes actually deteriorate or does the bound just become loose? It is shown that the BCH bound gives the true minimum distance within a factor of two. We conclude that it is the codes, not the bound, which deteriorate with increasing code length.