Lagrange identity for polynomials and $\delta$-codes of lengths $7t$ and $13t$

Abstract:

It is known that application of the Lagrange identity for polynomials (see [1]) is the key to composing four-symbol $\delta$-codes of length $(2s + 1)t$ for $s = {2^a}{10^b}{26^c}$ and odd $t \leqslant 59$ or $t = {2^d}{10^e}{26^f} + 1$, where $a$, $b$, $c$, $d$, $e$ and $f$ are nonnegative integers. Applications of the Lagrange identity also lead to constructions of four-symbol $\delta$-codes of length $u$ for $u = 7t$ or $13t$. Consequently, new families of Hadamard matrices of orders $4uw$ and $20uw$ can be constructed, where $w$ is the order of Williamson matrices. Related topics on zero correlation codes are also discussed.