Q-ary Alphabet Research Articles

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.

On the Generalized Concatenated Construction for Codes in $${L_1}$$ and Lee Metrics

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2m we construct codes over $$\mathbb{Z}_4$$ with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2m+1 (with code distance 4). We construct codes over $$\mathbb{Z}_4$$ of length n with Lee distance n which under the Gray mapping yield Hadamard matrices of order 2n (under the additional condition that an Hadamard matrix of order n exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.

Efficient Encoding/Decoding of GC-Balanced Codes Correcting Tandem Duplications

Tandem duplication is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain et al. (2017) proposed the study of codes that correct tandem duplications. All code constructions are based on irreducible words. Such code constructions are almost optimal to combat tandem duplications of length at most k where k ≤ 3. However, the problem of designing efficient encoder/decoder for such codes has not been investigated. In addition, the method cannot be extended to deal with the case of arbitrary k, where k ≥ 4. In this work, we study efficient encoding/decoding methods for irreducible words over general q-ary alphabet. Our methods provide the first known efficient encoder/decoder for q-ary codes correcting tandem duplications of length at most k, where k ≤ 3. In particular, we describe an (1, m)-finite state encoder and show that when m = Θ(1/ε) and ϊ = Θ(1/ε), the encoder achieves rate that is ε away from the optimal rate. We also provide ranking/unranking algorithms for irreducible words and modify the algorithms to reduce the space requirements for the finite state encoder. Over the DNA alphabet (or quaternary alphabet), we also impose weight constraint on the codewords. In particular, a quaternary word is GC-balanced if exactly half of the symbols of are either C or G. Via a modification of Knuth's balancing technique, we provide an efficient method that translates quaternary messages into GC-balanced codewords and the resulting codebook is able to correct tandem duplications of length at most k, where k ≤ 3. In addition, we provide the first known construction of codes to combat tandem duplications of length at most k, where k ≥ 4. Such codes can correct duplication errors in linear-time and they are almost optimal in terms of rate.

A 2D non-overlapping code over a q-ary alphabet

We define a set of matrices over a finite alphabet where all possible overlaps between any two matrices are forbidden. The set is also enumerated by providing some recurrences counting particular classes of restricted words. Moreover, we analyze the asymptotic cardinality of the set according to the parameters related to the construction of the matrices.

List decoding for a multiple access hyperchannel

We obtain bounds on the rate of (optimal) list-decoding codes with a fixed list size L ≥ 1 for a q-ary multiple access hyperchannel (MAHC) with s ≥ 2 inputs and one output. By definition, an output signal of this channel is the set of symbols of a q-ary alphabet that occur in at least one of the s input signals. For example, in the case of a binary MAHC, where q = 2, an output signal takes values in the ternary alphabet {0, 1, {0, 1}}; namely, it equals 0 (1) if all the s input signals are 0 (1) and equals {0, 1} otherwise. Previously, upper and lower bounds on the code rate for a q-ary MAHC were studied for L ≥ 1 and q = 2, and also for the nonbinary case q ≥ 3 for L = 1 only, i.e., for so-called frameproof codes. Constructing upper and lower bounds on the rate for the general case of L ≥ 1 and q ≥ 2 in the present paper is based on a substantial development of methods that we designed earlier for the classical binary disjunctive multiple access channel.

An Improvement to Levenshtein's Upper Bound on the Cardinality of Deletion Correcting Codes

We consider deletion correcting codes over a q-ary alphabet. It is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this paper, only the bounds corresponding to the all-insertion case and the all-deletion case were known. We recover these as special cases. The bound from the all-deletion case, due to Levenshtein, has been the best known for more than 45 years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size.

Nonasymptotic Upper Bounds for Deletion Correcting Codes

Explicit nonasymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for q-ary alphabet and string length is shown to be of size at most (qn-q)/{(q-1)(n-1)}. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The nonasymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known nonasymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.

Finding passwords by random walks: how long does it take?

We compare the efficiency of a deterministic ‘lawnmower’ and random search strategies for finding a prescribed sequence of letters (a password) of length M in which all letters are taken from the same Q-ary alphabet. We show that, at best, a random search takes two times longer than a ‘lawnmower’ search.

On fragments of words over a q-ary alphabet

We consider several combinatorial problems concerning fragments of words over a q-ary alphabet. We obtain a number of new facts, some of them being previously known in the binary case only.

SIMPLE FAULT-TOLERANT ENCODING OVER q-ARY CSS QUANTUM CODES

CSS codes are a subfamily of stabilizer codes especially appropriate for fault-tolerant quantum computations. A very simple method is proposed to encode a general qudit when a Calderbank–Shor–Steane quantum code, defined over a q-ary alphabet, is used.

Polynomial Time Low-Density Parity-Check Codes With Rates Very Close to the Capacity of the $q$-ary Random Deletion Channel for Large $q$

We demonstrate polynomial-time deletion codes based on the verification-based decoding paradigm that come arbitrarily close to capacity. The random deletion channel takes n symbols from a q-ary alphabet, and each symbol is deleted independently with probability p. Taking advantage of recent improvements on the results of Luby and Mitzenmacher for verification-based decoding by Shokrollahi and Wang, we show how to design for any epsi>0 and sufficiently large n and q deletion codes with the following properties: the rate is (1-p)(1-epsi), the failure probability is n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1/epsi2)</sup> /q, and the computational complexity for encoding and decoding is n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1/epsi2)</sup> log q. We also extend these schemes to obtain the same results even if the undeleted symbols are also transposed arbitrarily, and if a sufficiently small number of random symbols are inserted

A class of I.P.P. codes with efficient identification

Let C be a code of length n over a q-ary alphabet. An n-word y is called a descendant of a set of t codewords x 1,…, x t if y i ∈{ x i 1,…, x i t } for all i=1,…, n. A code is said to have the t- identifying parent property ( t-i.p.p.) if for any n-word y that is a descendant of at most t parents it is possible to identify at least one of them. An explicit construction is presented of t-i.p.p. codes of rate bounded away from zero, for which identification can be accomplished with complexity poly (n) .

Beyond the orthogonal array concept

Consider a block code Γ of length n, over a q-ary alphabet. For a given integer t∈[0, n], the regularity of Γ at level t is measured by the dimension of a well-defined subspace F of R t+1 , called the t- form space of the code. In particular, Γ is an orthogonal array of strength t if and only if F= R t+1 . A central result of the paper is the close connection that exists between t-forms and T- designs in the Hamming scheme H( n, q). The t-form space F is shown to have the structure of an ideal, and the canonical generator of F to be computable from the distance distribution of the code Γ.

On the subword complexity of iteratively generated infinite words

We obtain a formula for the subword complexity of some infinite words obtained from the initial symbol 0 by arbitrarily substituting symbols 0,1,…, q−1 to fixed finite words of equal length in the q-ary alphabet. The result of the paper is a generalization of the formula for the subword complexity of the Thue–Morse word in Avgustinovich (Discrete Anal. Oper. Res. 1 (1994) 3).