Three classes of partitioned difference families and their optimal constant composition codes

Shanding Xu; Longjiang Qu; Xiwang Cao

doi:10.3934/amc.2020120

Article Contents

2022, Volume 16, Issue 3: 465-483. Doi: 10.3934/amc.2020120

This issue Previous Article New discrete logarithm computation for the medium prime case using the function field sieve Next Article Nearly optimal codebooks from generalized Boolean bent functions over

$ \mathbb{Z}_{4} $

Three classes of partitioned difference families and their optimal constant composition codes

1.
College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China
2.
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China
3.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received: January 31, 2020

Revised: July 31, 2020

Early access: December 2020

Published: August 2022

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Abstract

Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on $ v $ prime or composite, cyclotomy on a residue class ring $ {\mathbb{Z}}_{v} $ can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order $ e $ on the ring $ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $, where $ q_i $ and $ q_j $ ($ i\neq j $) may not be co-prime, which includes classical cyclotomy as a special case. Here, $ q_1 $, $ q_2 $, $ \cdots $, $ q_k $ are powers of primes with an integer $ e|(q_i-1) $ for any $ 1\leq i\leq k $. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and $ d $-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.

Keywords:

Mathematics Subject Classification: 11T22; 14G50.

Citation:

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Table 1. $ (A, K, \lambda) $ PDF constructed in this paper

$ A $	$ K $	$ \lambda $	Constraints	Ref.
$ R\times {\mathbb{Z}}_{e} $	$ [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}] $	$ e-2 $	$ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)\|(q_i-1) $ for $ 1\leq i\leq k $	Theorem 3.4
$ R $	$ [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}] $	$ \frac{e-1}{2} $	$ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e\|(q_i-1) $ for $ 1\leq i\leq k $	Theorem 3.6
$ {\mathbb{Z}}_{\frac{q^m-1}{e}} \times {\mathbb{Z}}_{k} $	$ [k\frac{q^{m-1}-1}{e}^{1}1^{k \frac{q^m-q^{m-1}}{e}}] $	$ k \frac{q^{m-2}-1}{e} $	$ e \|(q-1), \operatorname{gcd}(e, m)=1 $, $ 1 \leq k \leq e $, $ m>2 $	Theorem 3.9

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Table 2. Some optimal CCCs with parameters $ (n, M, d, [\omega_0, \omega_1, \cdots, \omega_{m-1}])_m $ from our PDFs

Parameters	Constraints
$ (e v, e v, e v-e+2, [{(e-1)}^{\frac{ev-1}{e-1}}1^{1}])_\frac{ev+e-2}{e-1} $	$ v=q_{1} q_{2} \cdots q_{k} $, $ e(e-1)\|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(v, v, v-\frac{e-1}{2}, [e^{\frac{v-1}{2e}}1^{\frac{v+1}{2}}]\right)_{\frac{v-1}{2 e}+\frac{v+1}{2}} $	$ v=q_{1} q_{2} \cdots q_{k} $, $ e\geq 3 $ is odd such that $ e\|(q_i-1) $ for $ 1\leq i\leq k $
$ \left(k \frac{q^{m}-1}{e}, k \frac{q^{m}-1}{e}, k \frac{q^{m-2}-1}{e}, [1^{k \frac{q^m-q^{m-1}}{e}} k\frac{q^{m-1}-1}{e}^{1}]\right)_{k \frac{q^m-q^{m-1}}{e}+1} $	$ e\|(q-1), \gcd(e, m)=1, $ $ 1\leq k\leq e, m>2 $

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