New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences

In an array over an alphabet of $v$ symbols, a $t$-set of column indices $\left\{c_1,\dots ,c_t\right\}$ is covered if each $t$-tuple of the alphabet occurs at least once as a row of the sub-array indexed by $c_1,\dots ,c_t$. A covering array, denoted by CA$\left(N;t,k,v\right)$, is an $N\times k$ array over an alphabet with $v$ symbols with the property that any $t$-set of columns is covered. Here, $N$ is the size and $t$ is the strength of the covering array. Raaphorst et al. (Des. Codes Cryptogr. (2014) 73:949-968) give a construction for a CA$\left(2q^3-1;3,q^2+q+1,q\right)$, which we denote as $R_q$, by using linear feedback shift register (LFSR) sequences with characteristic polynomial being a primitive polynomial over $F_q$. The array $R_q$ corresponds to a covering perfect hash family. We give a construction of covering arrays of strength 3 based on horizontally concatenating $x$ copies of $R_q$, for any prime power $q$ and $x\in \left\{2,q,q+1,q^2,q^2-q+1\right\}$. The coverage is completed by developing Roux-type constructions that exploit the structure of $R_q$ and remove repeated rows. Some of these covering arrays improve the previous best-known upper bound of the size $N$ of covering arrays with the same corresponding parameters.