Weight Codes Research Articles

A Lower Bound on the Probability of Undetected Error for Binary Constant Weight Codes

In this correspondence, we study the probability of undetected error for binary constant weight codes. First, we derive a new lower bound on the probability of undetected error for binary constant weight codes. Next, we show that this bound is tight if and only if the binary constant weight codes are generated from certain t-designs in combinatorial design theory. This means that these binary constant weight codes generated from certain t-designs are uniformly optimal for error detection. Along the way, we determine the distance distributions of such binary constant weight codes. In particular, it is shown that binary constant weight codes generated from Steiner systems are uniformly optimal for error detection. Thus, we prove a conjecture of Xia, Fu, Jiang, and Ling. Furthermore, the distance distribution of a binary constant weight code generated from a Steiner system is determined. Finally, we study the exponent of the probability of undetected error for binary constant weight codes. We derive some bounds on the exponent of the probability of undetected error for binary constant weight codes. These bounds enable us to extend the region in which the exponent of the probability of undetected error is exactly determined

Study on cipher propertys of constant weight codes

Constant weight code is an important error-correcting control code in communications. Basic structure of constant weight codes for some arriving at Johnson bound, A(n, 2u, w), is presented. Some correlative propertys of the codes, the solution of arriving at Johnson bound, and the results on the couple constant code and some constant weight codes are discussed. The conclusion is verified through four examples.

A New Table of Constant Weight Codes of Length Greater than 28

Existing tables of constant weight codes are mainly confined to codes of length $n\leq 28$. This paper presents tables of codes of lengths $29 \leq n \leq 63$. The motivation for creating these tables was their application to the generation of good sets of frequency hopping lists in radio networks. The complete generation of all relevant cases by a small number of algorithms is augmented in individual cases by miscellaneous constructions. These sometimes give a larger number of codewords than the algorithms.

Field programmable gate array (FPGA) for iterative code evaluation

Iterative codes such as low density parity check (LDPC) codes and turbo product codes (TPC) are of interest for data storage applications. To apply these codes to digital recording systems, two possible schemes are considered in this paper. In the first scheme, a single LDPC code of column weight j>2 is used to replace the Reed-Solomon (RS) code of the conventional channel. In the second scheme, an iterative code is used as the inner code and is concatenated with an outer RS code. Single parity check TPC code is considered for this scheme. We developed a high-throughput field programmable gate array (FPGA) platform to evaluate the error floor performance of LDPC codes and the error statistics of TPC codes in partial response (PR) channel with turbo equalization. High rate codes (rate 8/9 for LDPC code and rate 0.935 for TPC code) are evaluated for magnetic recoding application. The bit error rate (BER) performance of LDPC code down to 10/sup -10/ can be reached within 2 h using the FPGA platform. The TPC code error statistics are evaluated using about 10/sup 11/ bits at different signal-to-noise (SNR) levels. For practical implementation complexity and power consumption, 2 channel iterations and 2 TPC decoder iterations are employed. The results show that the gain by applying TPC code under 2 channel iterations and 2 TPC decoder iterations is marginal.

Properties of m-Sequence and Construction of Constant Weight Codes

In the letter, properties of m-sequence are derived, based on these properties, a family of binary nonlinear constant weight codes is presented, these binary nonlinear constant weight codes can apply to automatic repeat request (ARQ) communication system, as detecting-error codes.

On constant composition codes

A constant composition code over a k-ary alphabet has the property that the numbers of occurrences of the k symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. In this paper, direct and recursive methods are developed for the construction of constant composition codes.

Optimal holey packings OHP4(2, 4, n ,3)'s

AbstractAn optimal holey packing OHPd(2, k, n, g) is equivalent to a maximal (g + 1)‐ary (n, k, d) constant weight code. In this paper, we provide some recursive constructions for OHPd(2, k, n, g)'s and use them to investigate the existence of an OHP4(2, 4, n, 3) for n ≡ 2, 3 (mod 4). Combining this with Wu's result (18), we prove that the necessary condition for the existence of an OHP4(2, 4, n, 3), namely, n ≥ 5 is also sufficient, except for n ∈ {6, 7} and except possibly for n = 26. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 111–123, 2006

New Binary Constant Weight Codes Based on Cayley Graphs of Groups and Their Decoding Methods

We propose a new class of binary nonlinear codes of constant weights derived from a permutation representation of a group that is given by a combinatorial definition such as Cayley graphs of a group. These codes are constructed by the following direct interpretation method from a group: (1) take one discrete group whose elements are defined by generators and their relations, such as those in the form of Cayley graphs; and (2) embedding the group into a binary space using some of their permutation representations by providing the generators with realization of permutations of some terms. The proposed codes are endowed with some good characteristics as follows: (a) we can easily learn information about the distances of the obtained codes, and moreover, (b) we can establish a decoding method for them that can correct random errors whose distances from code words are less than half of the minimum distances achieved using only parity checking procedures.

A Construction of Binary Constant-Weight Codes From Algebraic Curves Over Finite Fields

By employing the narrow ray class groups of algebraic curves, we give a construction of constant weight codes. This construction is a generalization of the one proposed by Xing. It turns out that this generalization gives an improvement on the lower bound of binary constant codes in the earlier work of Xing, while the latter one improves an earlier result of Graham and Sloane.

Existence of Generalized Steiner Systems GS(2,4,ν,2)

Generalized Steiner systems GS(2,4,v,2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance five, in which each codeword has length v and weight four. Etzion conjectured its existence for any integer v? 7 and v? 1(mod 3). The conjecture has been verified for prime powers v > 7 and v? 7(mod 12) by the latter two of the present authors. It has also been pointed out that there does not exist a GS(2,4,7,2). In this paper, constructions using frame generalized Steiner systems, two holey perfect bases and orthogonal Latin squares are discussed. With these constructions the conjecture is confirmed with the exception for v=7 and three possible exceptions for v ?13, 52, 58.

The construction of a 4EC-AUED code

In this paper, we construct a constant weight code based on the idea of Bose and Rao [2] and M. C. Lin [6]. This code can be used for correcting 4 symmetric errrors and simultaneously detecting all unidirectional errors. The code has size 70 and the information rate is 0.27.

Some new optimal quaternary constant weight codes

Constant weight codes (CWCs) are an important class of codes in coding theory. Generalized Steiner systems GS(2, k , v , g ) were first introduced by Etzion and used to construct optimal nonlinear CWCs over an alphabet of size g +1 with minimum Hamming distance 2 k −3, in which each codeword has length v and weight k . In this paper, Weil's theorem on character sum estimates is used to show that there exists a GS(2,4, v ,3) for any prime v ≡ 1 ( mod 4) and v >13. From the coding theory point of view, an optimal nonlinear quaternary ( v ,5,4) CWC exists for such a prime v .

Error-Correcting Codes and<tex>$B_h$</tex>-Sequences

We construct error-correcting (nonlinear) binary codes using a construction of Bose and Chowla in additive number theory. Our method extends a construction of Graham and Sloane for constant weight codes. The new codes improve 1028 of the 7168 best known h-error-correcting codes of word length /spl les/512 with 1/spl les/h/spl les/14. We give asymptotical comparisons to shortened Bose-Chaudhuri-Hocquenghem (BCH) codes.

New 2004 CPT Codes

New 2004 <i>CPT</i> Codes

Subspecialty Corner

Subspecialty Corner

Transmission time analysis for the parallel asynchronous communication scheme

In asynchronous systems, the sender encodes a data word with a code word from an unordered code and transmits the code word on the parallel bus lines. In this paper, a transmission time analysis for the above parallel asynchronous communication scheme is presented. It is proven that the average transmission time for a code word is a strictly increasing function of the weight of the code word and it approaches the worst transmission time possible when the weight goes to infinity. This implies that fast parallel asynchronous systems can be designed using low weight codes. This paper also analyzes the transmission time performances of the proximity detecting codes and gives some efficient low constant weight code designs.

Comparison of diverse optical CDMA codes using a normalized throughput metric

A new method of comparing optical CDMA codes of different families, sizes and weights is described. We outline why the traditional performance metric of bit-error rate versus number of simultaneous users is lacking and propose a new performance measure - the peak throughput normalized with respect to the size of the code. This new metric is used to show that optical-orthogonal codes (OOCs) with a weight of 4 perform best at low offered loads while OOCs with weight 5 should be used at higher offered loads. By applying the technique across different families of codes, we demonstrate that multi-wavelength OOCs (MWOOCs) perform better than both OOCs (by a factor of approximately 1.25) and asymmetric prime-hop codes (by a factor of approximately 3.5), over a wide range of offered loads.

The extremal codes of lengths 76 with an automorphism of order 19

We find all extremal [76,38,14] binary self-dual codes having automorphism of order 19. There are three inequivalent such codes. One of them was previously known. The other two are new. These codes are the shortest known self-dual codes of minimal weight 14 as well as the best-known linear codes of that length and dimension.

On equidistant constant weight codes

Equidistant constant weight codes are studied in this paper. The dual distance distribution of equidistant constant weight codes is investigated and used to obtain upper bounds on the size of such codes as well as equidistant codes in general.

Several classes of (2 m−1, w,2) optical orthogonal codes

Optical orthogonal codes have applications in optical code-division multiple access communication systems and other wide band code-division multiple environments. They can also be used to construct protocol sequences for multiuser collision channel without feedback, and constant weight codes for error detection and correction. In this paper, we construct four classes of (2 m −1, w,2) cyclotomic optical orthogonal codes, where w∈{5,9} or w⩾11 is any prime having primitive root 2, and therefore the corresponding classes of binary constant-weight cyclic codes.