Construction Of New Codes Research Articles

Two classes of optimal [formula omitted]-ary few-weight codes from down-sets

Recently, several infinite families of minimal and optimal linear codes are obtained from simplicial complexes or down-sets by Hyun and Wu et al. Motivated by their work, we give two new constructions of codes over the ring Fp+uFp,u2=u by using down-sets. We compute the Lee weight distributions of two classes of codes when the down-sets are generated by a single maximal element. Moreover, by using the Gray map, we obtain two classes of codes, which are minimal and distance optimal.

Construction of new codes in term‐rank metric

This study investigates codes in the term-rank metric. These codes can be used to correct crisscross errors in a ( M × N ) matrix. These errors can be found in memory chip arrays, in magnetic tape recordings, or in the parallel-channel system with interference. In this study, the authors will build an algorithm for computing term-rank of a matrix and new construction of term-rank codes. Single error-correcting codes in the term-rank metric are considered and help us build a generator matrix and parity-check matrix for term-rank codes. Research results derived from this study show that the term-rank metric is suitable for correcting crisscross errors.

Optimal Few-Weight Codes From Simplicial Complexes

Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\Bbb F_2+u\Bbb F_2$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.

Harnessing oversampling in correlation-coded OTDR.

Coding technique has long been investigated as a solution to improve the signal-to-noise ratio (SNR) without sacrificing spatial resolution in optical time domain reflectometry (OTDR) systems. The past researches have been focusing at the construction of new codes, the combination of coding and other techniques, and the application of coding technique to versatile distributed optical fiber sensing systems such as Raman OTDR and Brillouin optical time domain analyzer, where the results are fruitful. Here, we reveal that oversampling after photodetection opens up a new dimension for coded OTDR other than code length and code type. We demonstrate that the coding gain can be further improved by harnessing the oversampling. Furthermore, the photodetector's bandwidth-limited feature can also be used to select the optimal sampling rate in order to obtain additional SNR enhancement. We believe this principle could be applied to any practical correlation-coded OTDR-based distributed fiber sensing systems with sufficient SNR enhancement. Our findings can serve to update existing instruments based on correlation-coded OTDR in a straightforward manner and at a relatively low cost.

Construction of New Codes from Given Ones in an Additive Channel

In the present work, a construction making possible creation of an additive channel of cardinality s and rank r for arbitrary integers s, r, n (r≤min (n,s-1)), as well as creation of a code correcting errors of the channel A is presented.

Subspace Codes Based on Graph Matchings, Ferrers Diagrams, and Pending Blocks

This paper provides new constructions and lower bounds for subspace codes, using Ferrers diagram rank-metric codes from matchings of the complete graph and pending blocks. We present different constructions for constant dimension codes with minimum injection distance 2 or k - 1, where k is the constant dimension. Furthermore, we present a construction of new codes from old codes for any minimum distance. Then, we construct nonconstant dimension codes from these codes. Some examples of codes obtained by these constructions are the largest known codes for the given parameters.

New Constructions of Codes for Asymmetric Channels via Concatenation

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. In the binary case, Varshamov has shown that almost all good linear codes for the asymmetric channel are also good for the symmetric channel. Our results indicate that Varshamov's argument does not extend to the nonbinary case, i.e., one can find better linear codes for asymmetric channels than for symmetric ones.

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

A binary nonlinear code can be represented as a union of cosets of a binary linear subcode. In this paper, the complexity of some algorithms to obtain this representation is analyzed. Moreover, some properties and constructions of new codes from given ones in terms of this representation are described. Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes. All results are written in such a way that they can be easily transformed into algorithms, and the performance of these algorithms is evaluated.

Construction of new codes and its related parameters

In this correspondence, we have constructed new codes and discussed its parameters. We have related the covering radius and normality of these codes and discussed various concept of the codes. In particular, we have given the exact minimum distance and covering radius.

New constructions of codes meeting the Griesmer bound

For any binary linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">( n , k , d)</tex> code the Grfesmer bound says that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geq \sum_{i=0}^{k-1} \lceil d/2^{i} \rceil</tex> . We investigate codes that meet this bound with equality. We give new descriptions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\sum_{i=0}^{k-1}\lceil d/2^{i} \rceil , k, d)</tex> codes that have earlier been constructed by Solomon and Stiffler, Belov, and Helleseth and van Tilborg. Finally we show how to construct several new families of such codes with parameters not obtainable by any previous known constructions.

A survey of constructive coding theory, and a table of binary codes of highest known rate

Although more than twenty years have passed since the appearance of Shannon's papers, a still unsolved problem of coding theory is to construct block codes which attain a low probability of error at rates close to capacity. However, for moderate block lenghts many good codes are known, the best-known being the BCH codes discovered in 1959. This paper is a survey of results in coding theory obtained since the appearance of Berlekamp's “Algebraic coding theory” (1968), concentrating on those which lead to the construction of new codes. The paper concludes with a table giving the smallest redundancy of any binary code, linear or nonlinear, that is presently known (to the author), for all lengths up to 512 and all minimum distances up to 30.