arXiv subjects

Maheshanand Bhaintwal

Publications and source records attributed to Maheshanand Bhaintwal.

10 recordsLinked to original sources

arXiv2026

Maximal achievable service rates of some classes of linear codes

In this paper, we investigate lower bounds on the maximum achievable service rates for data symbols in certain classes of linear codes, including cyclic codes and low-density parity-check (LDPC) codes, that are derived from combinatorial structures such as $t$-designs, difference sets, and balanced incomplete block designs (BIBDs). We first establish a lower bound on the maximum achievable service rate for each data symbol in the following two cases: (i) systematic linear codes $C$ under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a BIBD, and (ii) non-systematic binary codes under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a $t$-design. We then investigate the linear codes obtained from the incidence matrices of BIBDs, particularly certain classes of BIBD-LDPC codes, and show how the parameters of the underlying designs can be exploited to determine lower bounds on the maximum achievable service rates for the data symbols of the corresponding linear code. In addition, we analyze the maximum achievable service rates of systematic extended linear codes. We show that the existence of a symmetric BIBD (SBIBD) corresponding to a dual codeword can be used to derive a lower bound on the maximum achievable service rate of the associated systematic cyclic code. We also present some families of cyclic codes constructed from difference sets and obtain explicit lower bounds on the maximum achievable service rates for their data symbols. In particular, we determine the exact values of the maximum achievable service rates for each data symbol of cyclic codes arising from Singer difference sets.

cs.IT↗

arXiv2026

Constructions of Quantum $(r,δ)$-LRCs from cyclic codes

Classical $(r,δ)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,δ)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,δ)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.

cs.IT↗

arXiv2025

The Service Rate Region of Hamming Codes

The service rate region of a coded distributed storage system is the set of all achievable data access requests under the capacity constraints. This paper investigates the service rate regions of systematic Hamming codes using hypergraph theory and derives bounds for the maximal achievable service rate of individual data objects. We establish upper bounds on the sum of service rates of data symbols indexed by a subset of systematic nodes in a systematic binary Hamming code, and explore the achievability of these bounds. Additionally, for non-systematic binary Hamming codes, we conclude that the aggregate service rate is limited by the number of columns of odd weight in the associated generator matrix.

cs.IT↗

arXiv2025

On equidistant single-orbit cyclic and quasi-cyclic subspace codes

A code is said to be equidistant if the distance between any two distinct codewords of the code is the same. In this paper, we have studied equidistant single-orbit cyclic and quasi-cyclic subspace codes. The orbit code generated by a subspace $U$ in $\mathbb{F}_{q^n}$ such that the dimension of $U$ over $\mathbb{F}_q$ is $t$ or $n-t$, $\mbox{where}~t=\dim_{\mathbb{F}_q}(\mbox{Stab}(U)\cup\{0\})$, is equidistant and is termed a trivial equidistant orbit code. Using the concept of cyclic difference sets, we have proved that only the trivial equidistant single-orbit cyclic subspace codes exist. Further, we have explored equidistant single-orbit quasi-cyclic subspace codes, focusing specifically on those which are sunflowers.

cs.IT↗

arXiv2024

On the distance distributions of single-orbit cyclic subspace codes

{A cyclic subspace code is a union of the orbits of subspaces contained in it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89, 447-470, 2021) showed that the study of the distance distribution of a single orbit cyclic subspace code is equivalent to the study of its intersection distribution. In this paper we have proved that in the orbit of a subspace $U$ of $\mathbb{F}_{q^n}$ that has the stabilizer $\mathbb{F}_{q^t}^*(t \neq n)$, the number of codeword pairs $(U,αU)$ such that $\dim(U\cap αU)=i$ for any $i,~ 0\leq i < \dim(U)$, is a multiple of $q^t(q^t+1)$, if $\frac{n}{t}$ is an odd number. In the case of even $\frac{n}{t}$, if $U$ contains $\frac{q^{2tm}-1}{q^{2t}-1}~ (m\geq 0)$ distinct cyclic shifts of $\mathbb{F}_{q^{2t}}$, then the number of codeword pairs $(U,αU)$ with intersection dimension $2tm$ is equal to $q^t+rq^t(q^t+1)$, for some non-negative integer $r$; and the number of codeword pairs $(U,αU)$ with intersection dimension $i,~(i\neq 2tm)$ is a multiple of $q^t(q^t+1)$. Some examples have been given to illustrate the results presented in the paper.

cs.IT↗

arXiv2022

A new class of negabent functions

Negabent functions were introduced as a generalization of bent functions, which have applications in coding theory and cryptography. In this paper, we have extended the notion of negabent functions to the functions defined from $\mathbb{Z}_q^n$ to $\mathbb{Z}_{2q}$ ($2q$-negabent), where $q \geq 2$ is a positive integer and $\mathbb{Z}_q$ is the ring of integers modulo $q$. For this, a new unitary transform (the nega-Hadamard transform) is introduced in the current set up, and some of its properties are discussed. Some results related to $2q$-negabent functions are presented. We present two constructions of $2q$-negabent functions. In the first construction, $2q$-negabent functions on $n$ variables are constructed when $q$ is an even positive integer. In the second construction, $2q$-negabent functions on two variables are constructed for arbitrary positive integer $q \ge 2$. Some examples of $2q$-negabent functions for different values of $q$ and $n$ are also presented.

cs.DM↗

arXiv2020

On cyclic LRC codes that are also LCD codes

Locally recoverable (LRC) codes provide a solution to single node failure in distributed storage systems, where it is a very common problem. On the other hand, linear complementary dual (LCD) codes are useful in fault injections attacks on storage systems. In this paper, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic LRC codes so that they are also LCD codes. A lower bound on the minimum distance of such codes is determined. Some examples have been given to explain the construction.

cs.IT↗

arXiv2017

One generator quasi-cyclic codes over F2 + uF2 + vF2 + uvF2

In this paper, we study the structure of 1-generator quasi-cyclic codes over the ring R = F2 + uF2 + vF2 + uvF2, with u2 = v2 = 0 and uv = vu. We determine the minimal spanning sets for these codes. As a generalization of these codes, we also investigate the structure of 1-generator generalized quasi-cyclic codes over R and determine a BCH type bound for them.

cs.IT↗

arXiv2015

Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

In this paper, we have studied cyclic codes over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over $R$ to be a $\mathbb{Z}_4$-free module is presented. We have provided the general form of the generators of a cyclic code over $R$ and determined a formula for the ranks of such codes. In this paper we have mainly focused on principally generated cyclic codes of odd length over $R$. We have determined a necessary condition and a sufficient condition for cyclic codes of odd lengths over $R$ to be $R$-free.

cs.IT↗

arXiv2014

Negacyclic codes over Z4+uZ4

In this paper, we study negacyclic codes of odd length and of length $2^k$ over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We give the complete structure of negacyclic codes for both the cases. We have obtained a minimal spanning set for negacyclic codes of odd lengths over $R$. A necessary and sufficient condition for negacyclic codes of odd lengths to be free is presented. We have determined the cardinality of negacyclic codes in each case. We have obtained the structure of the duals of negacyclic codes of length $2^k$ over $R$ and also characterized self-dual negacyclic codes of length $2^k$ over $R$.

cs.IT↗