On the average hitting times of weighted Cayley graphs

In the present paper, we give the exact formula for the average hitting time (HT, as an abbreviation) of random walks from one vertex to any other vertex on the weighted Cayley graph Cay($Z_{2n},\{\pm1\}$). Furthermore, we also give the exact formula for the HT's of random walks on the weighted Cayley graph Cay($Z_N,\{+1,+2\}$)

Jacobi polynomials and design theory II

In this paper, we introduce some new polynomials associated to linear codes over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We also give the notion of generalized $q$-colored $t$-designs. As an application of the generalized $q$-colored $t$-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over $\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.Comment: 28 page

A criterion for determining whether multiple shells support a tt-design

In this paper, we provide a criterion for determining whether multiple shells support a $t$-design. We construct as a corollary an infinite series of $2$-designs using power residue codes.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:2309.03206, arXiv:2305.03285, arXiv:2310.1428