Some new classes of 2-fold optimal or perfect splitting authentication codes

Abstract

Optimal restricted strong partially balanced t-design can be used to construct splitting authentication codes which achieve combinatorial lower bounds or information-theoretic lower bounds. In this paper, we investigate the existence of optimal restricted strong partially balanced 2-designs ORSPBD (v, k×c,1), and show that there exists an ORSPBD (v,2×c,1) for any positive integer v? v0 (mod 2c2) and v0?{1≤x≤2c2:gcd(x,c)=1orgcd(x,c)=c}?$v_{0}\in \{1\leq x\leq 2c^{2}:\ \gcd (x,c)=1\ \text {or} \ \gcd (x,c)=c \} \setminus ${c2+1≤x≤(c+1)2:gcd(x,c)=1andgcd(x,2)=2}$\{c^{2}+1\leq x\leq (c+1)^{2} :\gcd (x,c)=1\ \text {and}\ \gcd (x,2)=2\}$. Furthermore, we determine the existence of an ORSPBD (v,k×c,1) for any integer v?kc with (k,c)=(2,4), (2,5), (3,2) or for any even integer v?kc with (k,c)=(4,2). As their applications, we obtain six new infinite classes of 2-fold optimal or perfect c-splitting authentication codes.