Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: 11T22; 14G50.

 Citation:

• 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

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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