Maximum Number Of Codewords Research Articles

Optimal binary constant weight codes and affine linear groups over finite fields

The affine linear group of degree one, $$\text {AGL}(1,\mathbb {F}_q)$$ , over the finite field $$\mathbb {F}_q$$ , acts sharply two-transitively on $$\mathbb {F}_q$$ . Given $$S<\text {AGL}(1,\mathbb {F}_q)$$ and an integer k, $$1\le k\le q$$ , does there exist a k-element subset $$B\subset \mathbb {F}_q$$ whose set-wise stabilizer is S? Our main result is the derivation of two formulas which provide an answer to this question. This result allows us to determine all possible parameters of binary constant weight codes that are constructed from the action of $$\text {AGL}(1,\mathbb {F}_q)$$ on $$\mathbb {F}_q$$ to meet the Johnson bound. Consequently, for many parameters, we are able to determine the values of the function $$A_2(n,d,w)$$ , which is the maximum number of codewords in a binary constant weight code of length n, weight w and minimum distance $$\ge d$$ .

Construction of optimal constant-dimension subspace codes

A subspace code of length $$n$$n over the finite field $$\mathbb {F}_q$$Fq is a collection of subspaces of the $$n$$n-dimensional vector space $$\mathbb {F}_q^n$$Fqn. Subspace codes are applied to a number of areas such as noncoherent linear network coding and linear authentication. A challenge in the research of subspace codes is to construct large codes with prescribed code parameters, such that the codes have the maximum number of codewords, or the number of codewords is larger than that of previously known codes. In the literature, a general method was proposed for the construction of large constant-dimension subspace codes based on integer linear programming. In this work, making use of an optimization approach for finding the maximum independent set of a graph, a procedure is developed for constructing large subspace codes. The procedure, in some cases, outperforms the existing approach based on integer linear programming, and finds new subspace codes that have more codewords than existing codes.

Optimal equi-difference conflict-avoiding codes of weight four

A conflict-avoiding code (CAC) of length $$n$$n and weight $$w$$w is defined as a family $${\mathcal C}$$C of $$w$$w-subsets (called codewords) of $${\mathbb {Z}}_n$$Zn, the ring of residues modulo $$n$$n, such that $$\Delta (C) \cap \Delta (C') = \emptyset $$Δ(C)?Δ(C?)=? for any $$C, C' \in {\mathcal C}$$C,C??C, where $$\Delta (C) = \{ j-i \pmod {n} : i, j \in C, i \ne j\}$$Δ(C)={j-i(modn):i,j?C,i?j}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is called an equi-difference code if every codeword $$C \in {\mathcal C}$$C?C has the form $$\{ 0, i, 2i, \ldots , (w-1) i \}$${0,i,2i,?,(w-1)i}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is said to be optimal if $${\mathcal C}$$C has the maximum number of codewords. In this article, we investigate sizes and constructions of optimal codes in equi-difference CACs of weight four by using properly defined directed graphs. As a consequence, several series of infinite number of optimal equi-difference CACs are also provided.

Recursive Construction of n-gonal Codes on the Basis of Block Design

In the article are defined nonlinear n-gonal block codes. The methods of constructing n-gonal codes are considered. Suggest efficient universal recursive methods of constructing codes great length on the basis of block design for error-correcting code. On the basis of these methods can be constructed error-correcting codes for any predetermined number of errors. Among the codes with a predetermined length codeword and a predetermined number of units in the word, these codes will have a maximum number of codewords. Dignity of these codes is speed of encoding and decoding. Also possibility of fast change of a code without change of tables of encoding and decoding. This makes it possible to use these in cryptosystems.

Secure HARQ With Multiple Encoding Over Block Fading Channels: Channel Set Characterization and Outage Analysis

We consider a scenario where agent Alice aims at transmitting a message to agent Bob, while keeping it secret from a third eavesdropper agent Eve. The transmissions occur on block-fading channels, and Alice only has a statistical channel state information (CSI) on both Bob and Eve channels. In this paper, we add secrecy features to hybrid automatic repeat request (HARQ) in what becomes a secure HARQ (S-HARQ) scheme. In particular, Alice encodes the secret message and splits the codeword into blocks. Then, she transmits one block at a time and after each transmission Bob feeds back an acknowledge (ACK) (not acknowledge (NACK)) packet, indicating he was (was not) able to decode the secret message. Upon an ACK Alice stops transmission, while upon a NACK Alice re-encodes the secret message into a new codeword and transmits it block by block, waiting for a new feedback. The process is iterated until either Bob decodes the message or a maximum number of codewords has been transmitted. We characterize the set of channels for which there exists a sequence of codes that ensure both vanishing error probability to Bob and vanishing rate of information leakage to Eve in the limit of infinitely long codewords. We also analyze the secrecy and decodability outage probabilities, providing closed form expressions for the case of additive white Gaussian noise channels with Rayleigh fading. The tradeoff between the number of blocks per codeword and the maximum number of codewords to be transmitted is also discussed.

Heuristic Algorithms for Constructing Binary Constant Weight Codes

Constant weight binary codes are used in a number of applications. Constructions based on mathematical structure are known for many codes. However, heuristic constructions unrelated to any mathematical structure can become of greater importance when the parameters of the code are larger. This paper considers the problem of finding constant weight codes with the maximum number of codewords from a purely algorithmic perspective. A set of heuristic and metaheuristic methods is presented and developed into a variable neighborhood search framework. The proposed method is applied to 383 previously studied cases with lengths between 29 and 63. For these cases it generates 153 new codes, with significantly increased numbers of codewords in comparison with existing constructions. For 10 of these new codes the number of codewords meets a known upper bound, and so these 10 codes are optimal. As well as the ability to generate new best codes, the approach has the advantage that it is a single method capable of addressing many sets of parameters in a uniform way.

Construction of Constant GC-Content DNA Codes via a Variable Neighbourhood Search Algorithm

DNA codes are sets of words of fixed length n over the alphabet {A,C,G,T} which satisfy a number of combinatorial conditions. They have application in DNA computing, in DNA microarray technologies and as molecular bar codes. The combinatorial conditions considered are (i) minimum Hamming distance d, (ii) fixed GC content and, in some cases (iii) minimum distance d between any codeword and the reverse Watson-Crick complement of any codeword. The problem is to find DNA codes with the maximum number of codewords. In this paper the construction of DNA codes is studied from an algorithmic perspective. Four local search algorithms are developed and combined in a variable neighbourhood search framework. The algorithm has been run extensively. Over 254 problems considered, it was able to match or improve the best known lower bounds in 180 cases, with 52 new bests.

Some combinatorial constructions for optimal perfect deletion-correcting codes

There are two kinds of perfect t-deletion-correcting codes of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(k ? t, k, v)-codes and T(k ? t, k, v)-codes respectively. The cardinality of a T(k ? t, k, v)-code is determined by its parameters, while T*(k ? t, k, v)-codes do not necessarily have a fixed size. Let N(k ? t, k, v) denote the maximum number of codewords in any T*(k ? t, k, v)-code. A T*(k ? t, k, v)-code with N(k ? t, k, v) codewords is said to be optimal. In this paper, some combinatorial constructions for optimal T*(2, k, v)-codes are developed. Using these constructions, we are able to determine the values of N(2, 4, v) for all positive integers v. The values of N(2, 5, v) are also determined for almost all positive integers v, except for v = 13, 15, 19, 27 and 34.

Cyclically permutable representations of cyclic codes

Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple-access (CDMA) radio systems and optical CDMA. The usual emphasis is on finding constant weight cyclically permutable codes with the maximum number of codewords. In this paper the question of when a particular error-correcting code is equivalent (by permutation of the symbols) to a cyclically permutable code is addressed. The problem is introduced for simplex codes and a motivating example is given. In the final section it is shown that the construction technique may be applied in general to cyclic codes.

A Comment on “Systematic Single Asymmetric Error-Correcting Codes”

Bose and Al-Bassam designed quasi-systematic single asymmetric error-correcting codes that are able to encode k information bits in a systematic way, but cannot encode all 2/sup k/ information words. However, "Construction I" of their paper does not provide an efficient way for computing the number of codewords. In this correspondence, we derive a recursive relation based on generating functions in order to compute the maximum number of codewords in such codes. This recursion allows us to determine those numbers for large code lengths which may not be feasible otherwise.

On the Maximal Codes of Length 3 with the 2-Identifiable Parent Property

A q-ary code has identifiable parent property (IPP) if it allows one of the parents of a descendant word to be found. A 2-IPP code ensures that at least one parent of a pirate word constructed by a coalition of two users can be found. In this paper, we answer a question raised in [H. D. L. Hollmann et al., J. Combin. Theory Ser. A, 82 (1998), pp. 121--133] and show that F(q), the maximum number of codewords in a 2-IPP code of length 3, satisfies $|{\cal G}_0| \leq F(q) \leq |{\cal G}_0| +2$, where ${\cal G}_0$ is a well-defined graph. We also give an efficient algorithm (O(q3 )) for finding maximal codes.

New results on frame-proof codes and traceability schemes

We derive lower bounds on the maximum number of codewords in a class of frame-proof codes and traceability schemes, and give constructions for both with more codewords than the best known.

Optimal binary one-error-correcting codes of length 10 have 72 codewords

The maximum number of codewords in a binary code with length n and minimum distance d is denoted by A(n, d). By construction it is known that A(10, 3)/spl ges/72 and A(11, 3)/spl ges/144. These bounds have long been conjectured to be the exact values. This is here proved by classifying various codes of smaller length and lengthening these using backtracking and isomorphism rejection. There are 562 inequivalent codes attaining A(10, 3)=72 and 7398 inequivalent codes attaining A(11, 3)=144.

A new upper bound for binary codes with minimum distance four

The purpose of this paper is to give an upper bound for A[ n,4], the maximum number of codewords in a binary code of length n with minimum distance d between codewords.

New lower bounds for constant weight codes

Some new lower bounds are given for A(n,4,w), the maximum number of codewords in a binary code of length n, minimum distance 4, and constant weight w. In a number of cases the results significantly improve on the best bounds previously known. >

The equivalence of certain equidistant binary codes and symmetric bibds

We study equidistant codes of length 4k + 1 having (constant) weight 2k, and (constant) distance 2k between codewords. The maximum number of codewords is 4k; this can be attained if and only ifk = (u 2 +u)/2 (for some integeru) and there exists a ((2u 2 + 2u + 1,u 2, (u 2 −u)/2) — SBIBD. Also, one can construct such a code, with 4k − 1 codewords, from a (4k − 1, 2k − 1,k − 1) — SBIBD.

A lower bound for&amp;lt;tex&amp;gt;A(n,4,w)&amp;lt;/tex&amp;gt;(Corresp.)

An explicit lower bound is given for A(n,4,w) , the maximum number of codewords in a binary code of length n , minimum distance four, and constant weight w .

Bounds for binary codes just outside the plotkin range

Improved upper bounds for A ( n, d ), the maximum number of codewords in a binary code of word length n and minimum distance at least d , are presented in case n = 2 d + j , where j is small.

Bounds for binary codes of length less than 25

Improved bounds for A(n,d) , the maximum number of codewords in a (linear or nonlinear) binary code of word length n and minimum distance d , and for A(n,d,w) , the maximum number of binary vectors of length n , distance d , and constant weight w in the range n \leq 24 and d \leq 10 are presented. Some of the new values are A

Maximal Group Codes with Specified Minimum Distance

All n-digit maximal block codes with a specified minimum distance d such that 2d ≥ n can be constructed from the Hadamard matrices. These codes meet the Plotkin bound. In this paper we construct all maximal group codes in the region 2d ≥ n, where d is a specified minimum distance and n is the number of digits per code word. Unlike the case of block codes, the Plotkin upper limit, in general, fails to determine the number of code words B(n, d) in a maximal group code in the region 2d ≥ n. We show that the value of B(n, d) largely depends on the binary structure of the number d. An algorithm is developed that determines B(n, d), the maximum number of code words for given d and n ≤ 2d. The maximal code is, then, given by its modular representation, explicitly in terms of certain binary coefficients and constants related to n and d. As a side result, we obtain a new upper bound on the number of code words in the region 2d < n which is, in general, stronger than Plotkin's extended bound.