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.     

Odd weight symmetry in some binary codes (Corresp.)

If a linear binary code of lengthncontains the all-one codeword, than the weights of the code are symmetric. We consider those codes which do not contain the all-one codeword and yet have an equal number of symmetrically placed odd weight words.     

Further results on the synchronization of binary cyclic codes (Corresp.)

It is shown that certain coset codes derived from binary cyclic codes can determine the magnitude of a synchronization error, as well as its direction by examining only the syndrome of the receivedntuple. For such coset codes, therefore, the need for a search procedure to recover synchronism is eliminated. In addition, the range of slip that can be detected and corrected for noisy channels is extended.     

Detecting and correcting multiple bursts for binary cyclic codes (Corresp.)

Stone^1found that multiple-error-correcting codes inherently have the ability to correct multiple bursts. Using his methods, somewhat stronger theorems are derived here, and a decoding procedure is given.     

Equidistant binary arithmetic codes (Corresp.)

LetC(B)denote the binary cyclicANcode with generatorA, whereAB=2^{n} - 1. It is known thatC(B)is equidistant ifBis a prime powerp^{k}, where either2or-2is primitive moduloBprovidedpequiv 1 pmod{3}{rm if} k > 1. It is conjectured that these are the onlyBsuch thatC(B)is equidistant. We have verified this forB < 100 000. Several results are established that further limit the possibilities for counterexamples to the conjecture.     

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.     

Synchronization of binary source codes (Corresp.)

The problem of achieving synchronization for variable-length source codes is addressed through the use of self-synchronizing binary prefix-condition codes. Although our codes are suboptimal in the sense of minimum average codeword length, they have the advantages of being generated by an explicit constructive algorithm, having minimal additional redundancy compared with optimal codes-as little as one additional bit introduced into the least likely codeword for a large class of sources-and having statistical synchronizing performance that improves on that of the optimal code in many cases.     

On binary majority-logic decodable codes (Corresp.)

LetVprimebe a binary(n,k)majority-logic decodable code withgprime (X)as its generator polynomial and odd minimum distanced. LetVbe the(n, k - 1)subset code generated bygprime (X)(1 + X). This correspondence shows thatVis majority-logic deeodable withd + 1orthogonal estimates. This fact is useful in the simultaneous correction of random errors and erasures.     

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.     

Some efficient binary codes constructed using Srivastava codes (Corresp.)

In this correspondence, we construct a new class of binary codes by exploiting the symmetry properties of the parity check matrix of the Srivastava codes. The construction is a generalization of Goppa's construction [1]. A number of the binary codes constructed are proved equal, or superior, to the best codes previously known.