arXiv subjects

Minjia Shi

Publications and source records attributed to Minjia Shi.

At least 91 records · Page 5Linked to original sources

arXiv2019

On $Z_pZ_{p^k}$-additive codes and their duality

In this paper, two different Gray-like maps from $Z_p^α\times Z_{p^k}^β$, where $p$ is prime, to $Z_p^n$, $n={α+βp^{k-1}}$, denoted by $ϕ$ and $Φ$, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^\bot$ is its dual, then the weight enumerators of the image $p$-ary codes $ϕ(C)$ and $Φ(C^\bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given.

cs.IT↗

arXiv2018

Additive perfect codes in Doob graphs

The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$, where $n'+n''=n$. A set of vertices of $D(m,n)$ is called an additive code if it forms a subgroup of this group. We construct a $3$-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive $1$-perfect codes in $D(m,n'+n'')$ are sufficient. Additionally, two quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and $D(2667,0+127)$.

cs.IT↗

arXiv2018

How many weights can a cyclic code have ?

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $Γ(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$-ary, as well as in the $q$-ary Hamming code are determined, two difficult results of independent interest.

cs.IT↗

arXiv2018

A new distance-regular graph of diameter 3 on 1024 vertices

The dodecacode is a nonlinear additive quaternary code of length $12$. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance $5$. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on $2^{10}$ vertices, with new intersection array $\{33,30,15;1,2,15\}$. The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters $(2^{10},495,238,240)$. Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length $33$ is constructed.

math.CO↗

arXiv2018

On self-dual negacirculant codes of index two and four

In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely many such $p$'s exist that admit $q$ as a primitive root by the Artin conjecture in arithmetic progressions. This number theory conjecture is known to hold under GRH. We study the double (resp. four)-negacirculant codes over finite fields $\mathbb{F}_q, $ of co-index such $n$'s, including the exact enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound on the relative distance of the codes it contains.

cs.IT↗

arXiv2018

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. Güneri, H. Shoaib, P. Solé, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. Özdemir, P. Solé, 2017) and ( M. Shi, H. Zhu, P. Solé, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over $\F_q,$ for many classes of $q$'s. This is a partial solution to a fifty year old open problem.

cs.IT↗

arXiv2018

Optimal three-weight cubic codes

In this paper, we construct an infinite family of three-weight binary codes from linear codes over the ring $R=\mathbb{F}_2+v\mathbb{F}_2+v^2\mathbb{F}_2$, where $v^3=1.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distributions are computed by employing character sums. The three-weight binary linear codes which we construct are shown to be optimal when $m$ is odd and $m>1$. They are cubic, that is to say quasi-cyclic of co-index three. An application to secret sharing schemes is given.

cs.IT↗

arXiv2018

On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q + u\mathbb{F}_q$

Double circulant codes of length $2n$ over the semilocal ring $R = \mathbb{F}_q + u\mathbb{F}_q,\, u^2=u,$ are studied when $q$ is an odd prime power, and $-1$ is a square in $\mathbb{F}_q.$ Double negacirculant codes of length $2n$ are studied over $R$ when $n$ is even and $q$ is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length $2n$ is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length $4n$ over $\mathbb{F}_q$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.

cs.IT↗

arXiv2018

How many weights can a linear code have ?

We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general case when both $k$ and $q$ are $\ge 3.$ A refinement $L(n,k,q),$ as well as nonlinear analogues $N(M,q)$ and $N(n,M,q),$ are also introduced and studied.

cs.IT↗

arXiv2018

Linear codes with few weights over $\mathbb{F}_2+u\mathbb{F}_2$

In this paper, we construct an infinite family of five-weight codes from trace codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2$, where $u^2=0.$ The trace codes have the algebraic structure of abelian codes. Their Lee weight is computed by using character sums. Combined with Pless power moments and Newton's Identities, the weight distribution of the Gray image of trace codes was present. Their support structure is determined. An application to secret sharing schemes is given.

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arXiv2018

Double circulant self-dual and LCD codes over Galois rings

This paper investigates the existence, enumeration and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic $p^2$ and order $p^4$ with $p$ and odd prime. When $p \equiv 3 \pmod{4},$ we give an algorithm to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2},$ for the metric induced by the standard $\mathbb{F}_p$-valued Gray maps.

cs.IT↗

arXiv2017

On the proximity of large primes

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which differ from $m$ by at most one base $q$ digit.

math.NT↗

arXiv2017

On self-dual four circulant codes

Four circulant codes form a special class of $2$-generator, index $4$, quasi-cyclic codes. Under some conditions on their generator matrices they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert-Varshamov bound.

cs.IT↗

arXiv2017

Two-weight codes and second order recurrences

Cyclic codes of dimension $2$ over a finite field are shown to have at most two nonzero weights. This extends a construction of Rao et al (2010) and disproves a conjecture of Schmidt-White (2002). We compute their weight distribution, and give a condition on the roots of their check polynomials for them to be MDS.

cs.IT↗

arXiv2017

Two new families of two-weight codes

We construct two new infinite families of trace codes of dimension $2m$, over the ring $\mathbb{F}_p+u\mathbb{F}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1)^2$ and $2(p^m-1)^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\equiv 3 \pmod{4},$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3,$ or $p\ge 5$ and $m\ge 4.$ Applications to secret sharing schemes are given.

cs.IT↗

arXiv2017

Pisano period codes

The cyclic codes with parity check polynomial the reciprocal of the characteristic polynomial of the Fibonacci recurrence over a prime finite field are shown to have either one weight or two weights. When these codes are irreducible cyclic we obtain many counterexamples to the conjectural classification of two-weight irreducible cyclic codes of Schmidt and White (2002). When they are reducible and projective their duals are uniformly packed.

cs.IT↗

arXiv2017

Trace codes over $\Z_4$ and Boolean function

We construct trace codes over $\Z_4$ by using Boolean functions and skew sets, respectively. Their Lee weight distribution is studied by using a Galois ring version of the Walsh-Hadamard transform and exponential sums. We obtain a new family of optimal two-weight codes over $\Z_4.$

cs.IT↗

arXiv2017

Constructions of optimal LCD codes over large finite fields

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.

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