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

2017
*
arXiv
*
pre-print

In this paper, a generalized concatenation construction for q-

arXiv:1711.00189v1
fatcat:5xteig2s55ckpal4najhtkur4q
*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*...##
###
On full-rank perfect codes over finite fields

2016
*
Journal of Applied and Industrial Mathematics
*

We also present a generalization

doi:10.1134/s1990478916030157
fatcat:zlxzwuri2veghmoxj5yec2fhze
*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. ...##
###
On the number of q-ary quasi-perfect codes with covering radius 2
[article]

2021
*
arXiv
*
pre-print

In this paper we present a family

arXiv:2111.00774v1
fatcat:dqqcabupqbhmffxycmvzgex4mm
*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 . ...##
###
On the admissible families of components of Hamming codes
[article]

2012
*
arXiv
*
pre-print

It is shown that every q-

arXiv:1202.0349v1
fatcat:pj4ywoxsizd2vibqvi3zvklh6m
*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*). ...##
###
q-ary Propelinear Perfect Codes from the Regular Subgroups of the GA(r,q) and Their Ranks
[article]

2021
*
arXiv
*
pre-print

We propose a new method

arXiv:2112.08659v1
fatcat:65tc44cfxzhfpn7jjlvchnb2jq
*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+*1*−*1*is a q-*ary**perfect**code**of**length*q−*1*. ...##
###
On non-full-rank perfect codes over finite fields
[article]

2017
*
arXiv
*
pre-print

We show that the orthogonal

arXiv:1704.02627v1
fatcat:fjebxtb7ujetjghqcu7czj577m
*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] . ...##
###
An enumeration of 1-perfect ternary codes
[article]

2021
*
arXiv
*
pre-print

We enumerate ternary

arXiv:2110.06305v2
fatcat:xyz3nnlqrng25fdaeqbvy23b2e
*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−*1*−*1*)/2. ...##
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Resource placement in torus-based networks

1997
*
IEEE transactions on computers
*

This paper contains some solutions to

doi:10.1109/12.628393
fatcat:hxszykwnxvbxzeg46hlj6376q4
*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*)). ...##
###
M-Ary Mutually Orthogonal Complementary Gold Codes

2009
*
Zenodo
*

Publication in the conference proceedings

doi:10.5281/zenodo.41736
fatcat:s4kw6oixireq7onzvipokj7aqu
*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 . ...##
###
Every finite group is the automorphism group of some perfect code

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. ...

##
###
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

2015
*
Discrete Mathematics
*

We prove that every

doi:10.1016/j.disc.2015.04.014
fatcat:awilxjmxrzccnf6ybcsmpjip5u
*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*. ...##
###
Perfect (d,k)-codes capable of correcting single peak-shifts

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. ...

##
###
Ensuring Message Embedding in Wet Paper Steganography
[chapter]

2011
*
Lecture Notes in Computer Science
*

In this paper, we introduce a randomized syndrome

doi:10.1007/978-3-642-25516-8_15
fatcat:n6svbbv4rjajto5jlfh2eabwgm
*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. ...##
###
On q-ary shortened-1-perfect-like codes
[article]

2021
*
arXiv
*
pre-print

At second, we show the existence

arXiv:2110.05256v1
fatcat:bzyrhxeedzbexdfsjmfwmcln2a
*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). ...
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