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A generalized concatenation construction for q-ary 1-perfect codes [article]

Alexander M. Romanov
2017 arXiv   pre-print
In this paper, a generalized concatenation construction for q-ary 1-perfect codes is presented that allows us to construct q-ary 1-perfect codes of length (q - 1)nm + n + m from the given q-ary 1-perfect  ...  We consider perfect 1-error correcting codes over a finite field with q elements (briefly q-ary 1-perfect codes).  ...  Given a q-ary 1-perfect code C of length n, a q-ary 1-perfect code C ′ of length m, and a vector function f , defined on the set C with values in F m q , define the q-ary 1-perfect code F = {(x|c + P 1  ... 
arXiv:1711.00189v1 fatcat:5xteig2s55ckpal4najhtkur4q

On full-rank perfect codes over finite fields

A. M. Romanov
2016 Journal of Applied and Industrial Mathematics  
We also present a generalization of the Lindström and Schönheim construction of q-ary 1-perfect codes and provide a lower bound on the number of pairwise distinct q-ary 1-perfect codes of length n.  ...  Properties of i-components of q-ary Hamming codes are investigated and the construction of full-rank q-ary 1-perfect codes is based on these properties.  ...  provide a lower bound on the number of q-ary 1-perfect codes of length n.  ... 
doi:10.1134/s1990478916030157 fatcat:zlxzwuri2veghmoxj5yec2fhze

On the number of q-ary quasi-perfect codes with covering radius 2 [article]

Alexander M. Romanov
2021 arXiv   pre-print
In this paper we present a family of q-ary nonlinear quasi-perfect codes with covering radius 2.  ...  We prove that there are more than q^q^cn nonequivalent such codes of length n, for all sufficiently large n and a constant c = 1/q - ε.  ...  An arbitrary subset C of F n q is called a q-ary error correcting code (briefly a q-ary code). The length of a code C ⊆ F n q is the dimension of the vector space F n q .  ... 
arXiv:2111.00774v1 fatcat:dqqcabupqbhmffxycmvzgex4mm

On the admissible families of components of Hamming codes [article]

Alexander M. Romanov
2012 arXiv   pre-print
It is shown that every q-ary code of length m and minimum distance 5 (for q = 3 the minimum distance is 3) can be embedded in a q-ary 1-perfect code of length n = (q^m-1)/(q-1).  ...  It is also shown that every binary code of length m + k and minimum distance 3k + 3 can be embedded in a binary 1-perfect code of length n = 2^m-1.  ...  the strong sense) in the q-ary 1-perfect code T of length n = (q m − 1)/(q − 1).  ... 
arXiv:1202.0349v1 fatcat:pj4ywoxsizd2vibqvi3zvklh6m

q-ary Propelinear Perfect Codes from the Regular Subgroups of the GA(r,q) and Their Ranks [article]

Ivan Mogilnykh
2021 arXiv   pre-print
We propose a new method of constructing q-ary propelinear perfect codes.  ...  For any prime q it is shown that the new class contains an infinite series of q-ary propelinear perfect codes of varying ranks of growing length.  ...  a∈Fqr qr+11 is a q-ary perfect code of length q−1 .  ... 
arXiv:2112.08659v1 fatcat:65tc44cfxzhfpn7jjlvchnb2jq

On non-full-rank perfect codes over finite fields [article]

Alexander M. Romanov
2017 arXiv   pre-print
We show that the orthogonal code to the q-ary non-full-rank 1-perfect code of length n = (q^m-1)/(q-1) is a q-ary constant-weight code with Hamming weight equals to q^m - 1 where m is any natural number  ...  We suggest a generalization of the concatenation construction to the q-ary case and construct the ternary 1-perfect codes of length 13 and rank 12.  ...  For m = 3 and for q = p r , r > 1, the existence of full-rank q-ary 1-perfect codes is proved in [16] .  ... 
arXiv:1704.02627v1 fatcat:fjebxtb7ujetjghqcu7czj577m

An enumeration of 1-perfect ternary codes [article]

Minjia Shi
2021 arXiv   pre-print
We enumerate ternary 1-perfect codes of length 13 obtained by concatenation from codes of lengths 9 and 4; we find that there are 93241327 equivalence classes of such codes.  ...  We study codes with parameters of the ternary Hamming (n,3^n-m,3) code, i.e., ternary 1-perfect codes. The rank of the code is defined as the dimension of its affine span.  ...  Theorem 1. Let C be a 3-ary 1-perfect code of length n = (3 m − 1)/2 of rank at most +1 and C ⋆ be the 3-ary Hamming code of length n ′ = (3 m−11)/2.  ... 
arXiv:2110.06305v2 fatcat:xyz3nnlqrng25fdaeqbvy23b2e

Resource placement in torus-based networks

M.M. Bae
1997 IEEE transactions on computers  
This paper contains some solutions to perfect distance-t and perfect / quasi-perfect j-adjacency placement in a k-ary n-cube and a torus using Lee distance error-correcting codes.  ...  A vector c of length n over Z n k is a code word More on these codes will be described in Section 3.1. iff cH T = 0.  ...  From Theorem 1, C n p must satisfy (c mod p) H T = 0 where H is the check matrix of C n p constructed in the manner of the previous section, and c mod p = ((c 1 mod p) ( c 2 mod p) ( c n mod p)).  ... 
doi:10.1109/12.628393 fatcat:hxszykwnxvbxzeg46hlj6376q4

M-Ary Mutually Orthogonal Complementary Gold Codes

João Pereira, Henrique Silva
2009 Zenodo  
Publication in the conference proceedings of EUSIPCO, Glasgow, Scotland, 2009  ...  By using (4) and (5) , it is possible to generate any perfect sequence p x with length N whenever sequence x lies on a constant magnitude circle, for example ( ) x n N = , for 0 1 n N ≤ ≤ − .  ...  Let ( ) x n , with 0,1, 2..., 1 n N = − , be one of the N points of a periodic sequence x .  ... 
doi:10.5281/zenodo.41736 fatcat:s4kw6oixireq7onzvipokj7aqu

Every finite group is the automorphism group of some perfect code

K.T Phelps
1986 Journal of combinatorial theory. Series A  
perfect l-code of length (n + l)(m + 1).  ...  It is our purpose to establish this result for perfect l-codes. PRELIMINARIES A n-ary distance 2-code of length m + 1, having n"' codewords, is equivalent to an m-ary quasigroup of order n.  ...  In particular Professor Mendelsohn made some key observations which allowed for the relaxation of the hypothesis and greatly facilitated the proof of Lemma 4.1.  ... 
doi:10.1016/0097-3165(86)90021-x fatcat:maku7nuiwjd7xoujpypg5azb7m

Page 3828 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
(RS-AOSSI; Novosibirsk) On the structure of minimal distance graphs of perfect binary (n,3)-codes. (Russian. Russian summary) Diskretn. Anal. Issled. Oper. Ser. 1 5 (1998), no. 4, 3-5, 100.  ...  We first compute the linear programming bound on the dimension of such a code, then show that this bound can only be attained when the code either is of even length, or is of length 3 or 5.  ... 

Embedding in q-ary 1-perfect codes and partitions

Denis S. Krotov, Ev V. Sotnikova
2015 Discrete Mathematics  
We prove that every 1-error-correcting code over a finite field can be embedded in a 1-perfect code of some larger length.  ...  Keywords: error-correcting code, 1-perfect code, 1-perfect partition, embedding  ...  Introduction The goal of the current work is to show that any (in general, nonlinear) code that can correct at least one error is a subcode of a 1-perfect code of some larger length.  ... 
doi:10.1016/j.disc.2015.04.014 fatcat:awilxjmxrzccnf6ybcsmpjip5u

Perfect (d,k)-codes capable of correcting single peak-shifts

V.I. Levenshtein, A.J.H. Vinck
1993 IEEE Transactions on Information Theory  
Codes, consisting of sequences 0 OL 10 1 . . O a 1. where d 5 Q, 5 k . and call them (d,k)-codes of reduced length N are considered.  ...  Explicit constructions of such designs for t = 1, t = 2, and t = ( p -1)/2 are given, where p is a prime.  ...  This remark allows us to concentrate on (d,k)-codes of a fixed reduced length AV.  ... 
doi:10.1109/18.212300 fatcat:nosuyrxodff55hgrlnchpwcfxi

Ensuring Message Embedding in Wet Paper Steganography [chapter]

Daniel Augot, Morgan Barbier, Caroline Fontaine
2011 Lecture Notes in Computer Science  
In this paper, we introduce a randomized syndrome coding, which guarantees the embedding success with probability one. We analyze the parameters of this new scheme in the case of perfect codes.  ...  In 2005, Fridrich et al. introduced wet paper codes to improve the undetectability of the embedding by nabling the sender to lock some components of the cover-data, according to the nature of the cover-medium  ...  Consider a q-ary wet channel on length n with at most ℓ wet positions, and that there exists a q-ary code C whose dual code C ⊥ has parameters [n, k ⊥ , d ⊥ = ℓ] q with k ⊥ + d ⊥ = n + 1 − g.  ... 
doi:10.1007/978-3-642-25516-8_15 fatcat:n6svbbv4rjajto5jlfh2eabwgm

On q-ary shortened-1-perfect-like codes [article]

Minjia Shi
2021 arXiv   pre-print
At second, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.  ...  We study codes with parameters of q-ary shortened Hamming codes, i.e., (n=(q^m-q)/(q-1), q^n-m, 3)_q. At first, we prove the fact mentioned in [A.E.Brouwer et al.  ...  A q-ary code of length n is an arbitrary nonempty set of vertices of H(n, q).  ... 
arXiv:2110.05256v1 fatcat:bzyrhxeedzbexdfsjmfwmcln2a
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