On generator matrices of MDS codes (Corresp.)

It is shown that the family ofq-ary generalized Reed-Solomon codes is identical to the family ofq-ary linear codes generated by matrices of the form[I|A], whereIis the identity matrix, andAis a generalized Cauchy matrix. Using Cauchy matrices, a construction is shown of maximal triangular arrays over GF(q), which are constant along diagonals in a Hankel matrix fashion, and with the property that every square subarray is a nonsingular matrix. By taking rectangular subarrays of the described triangles, it is possible to construct generator matrices[I|A]of maximum distance separable codes, whereAis a Hankel matrix. The parameters of the codes are(n,k,d), for1 leq n leq q+ 1, 1 leq k leq n, andd=n-k+1.     

On synchronizable binary cyclic codes (Corresp.)

In this correspondence a method is presented whereby the average synchronization-error-correcting capability of Tavares' subset codes may be improved with no additional cost in rate and with only a small increase in the complexity of encoding and decoding. The method consists simply in shifting every word of the subset codes in such a way so that the shifted versions have a maximum number of leading and trailing zeros. A lower bound on the increase in synchronization-error-correcting capability provided by this method is derived.     

A bound for cyclic codes of composite length (Corresp.)

An upper bound on the minimum distance of cyclic codes of composite length is presented. This upper bound proves the BCH bound to be exact for many cyclic codes.     

Construction and decomposition of cyclic codes of composite length (Corresp.)

A construction of cyclic codes in which an "inner" cyclic code is concatenated with appropriately chosen "outer" cyclic codes is presented. An example of the construction that improves on BCH codes of low rate is provided. Conversely, any cyclic code of lengthn_1 n_2can be constructed by this construction, wheren_1andn_2are relatively prime.     

Low-density parity check codes over GF(q)

Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10^-5 at E_b/N₀=0.2 dB     

On the weight distribution of cyclic codes (Corresp.)

Letg(x)h(x) = x^n - 1, n = q^m - 1, and assume thath(x)contains a primitive factorf(x)of degreem. IfVis theq-ary(n, k)cyclic code generated byg(x), Uits subcode generated byg(x)f(x), then it will be shown that the weight distribution ofVcan be obtained from the weight distribution ofUand its cosetU + g(x).     

On BCH codes over arbitrary integer tings (Corresp.)

Bose-Chadhuri-Hocquenghem (BCH) codes with symbols from an arbitrary finite integer ring are derived in terms of their generator polynomials. Tile derivation is based on the factorization ofx^{n}-1over the unit ring of an appropriate extension of the Finite integer ring. The construction is thus shown to be similar to that for BCH codes over finite fields.     

Multiple-burst error-correcting cyclic product codes (Corresp.)

LetCbe the cyclic product code ofpsingle parity check codes of relatively prime lengthsn_{1}, n_{2},cdots , n_{p} (n_{1} < n_{2} < cdots < n_{p}). It is proven thatCcan correct2^{P-2}+2^{p-3}-1bursts of lengthn_{1}, andlfloor(max{p+1, min{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}}}-1)/2rfloorbursts of lengthn_{1}n_{2} cdots n_{s} (2leq s leq p-2). Forp=3this means thatCis double-burst-n_{1}-correcting. An efficient decoding algorithm is presented for this code.     

On the dimension and minimum distance of BCH codes over GF(q)

In this paper, only narrow-sense primitive BCH codes over GF(q) are considered. A formula, that can be used in many cases, is first presented for computing the dimension of BCH codes. It improves the result given by MacWilliams and Sloane in 1977. A new method for finding the dimension of all types of BCH codes is proposed. In second part, it is proved that the BCH bound is the leader of some cyclotomic coset, and we guess that the minimum distance for any BCH code is also the leader of some cyclotomic coset. Supported by the National Natural Science Foundation of China     

The covering radius of cyclic codes of length up to 31 (Corresp.)

The covering radius is given for all binary cyclic codes of length less than or equal to31. Many of these codes are optimal in the sense of having the smallest possible covering radius of any linear code of that length and dimension.     

Quantum codes from cyclic codes over GF(4m)

We provide a construction for quantum codes (Hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4^m). We also provide examples of these codes some of which meet the known bounds for quantum codes     

On the minimum distance of certain reversible cyclic codes (Corresp.)

The minimum distance of a class of reversible cyclic codes has been proved to be greater than that given by the BCH bound. It is also noted that this class of codes includes the class of primitive double-error-correcting binary codes of Melas as well as the class of nonprimitive double-error-correcting binary codes discovered by Zetterberg as special cases.     

Cycle representatives for minimal cyclic codes (Corresp.)

An algorithm is presented for the calculation of a cycle set for any cyclic code which has an irreducible parity-check polynomial.     

On simplex-type binary codes (Corresp.)

The rank of the generating matrix in certain cases is computed. The result is perhaps of interest in its own right, apart from any applications it may have. The method of proof, utilizing as it does a knowledge of the parameters of a certain association scheme, is a technique which can be profitably used elsewhere.     

Generation of m-sequences over GF(q2) from m-sequences over GF(q)

A closed form solution that yields the component shift required for the vector representation of m-sequences over GF(q²) in terms of m-sequences over GF(q²) is presented. Iterative application of this expression then enables the vctor representation of m-sequences over GF(q^2m) in terms of m-sequences over GF(q). These vector m-seqeences can be used directly for the selection of frequencies in frequency hopped spread spectrum communication systems.     

Characterization theorems for extending Goppa codes to cyclic codes (Corresp.)

Several theorems are presented which characterize Goppa codes having the property of becoming cyclic when an overall parity cheek is added. If such a Goppa code has location setL = GF (q^{m})and a Goppa polynomialg(z)that is irreducible overGF(q^{m}), theng(z)must be a quadratic. Goppa codes defined by(z- beta)^{a}and location setLwith cardinalitynsuch thatn+l|q^{m}-1are considered along with their subcodes. A sufficient condition onLis derived for the extended codes to become cyclic. This condition is also necessary whena= 1. The construction ofLfor differentnsatisfying the stated condition is investigated in some detail. Some irreversible Goppa codes have been shown to become cyclic when extended by an overall parity check.