Upper Bounds for the Lengths of $s$-Extremal Codes Over $BBF_{2}$, $BBF_{4}$, and $BBF_{2} + uBBF_{2}

Our purpose is to find an upper bound for the length of an $s$-extremal code over ${BBF }_{2}$ (resp. ${BBF}_{4}$) when $d equiv 2!!!pmod {4}$ (resp. $d$ odd). This question is left open in [A bound for certain $s$-extremal lattices and codes, Archiv der Mathematik, vol. 89, no.2, pp. 143-151, 2007] (resp. [$s$-extremal additive $BBF _{4}$ codes, Advances in Mathematics of Communications, vol. 1, no. 1, pp. 111-130,2007]). More precisely, we show that if there is an $[n, {{ n}over { 2}}, d]~s$-extremal Type I binary self-dual code with $d>6$ and $dequiv 2!!! pmod {4}$, then $n ... 21d-82$. Similarly we show that if there is an $(n, 2...{n}, d)~s$-extremal Type I additive self-dual code over $BBF _{4}$ with $d>1$ and $d equiv 1 pmod {2}$, then $n ... 13d-26$. We also define $s$-extremal self-dual codes over ${BBF}_{2} + u {BBF}_{2}$ and derive an upper bound for the length of an $s$-extremal self-dual code over ${BBF}_{2} + u {BBF}_{2}$ using the information on binary $s$-extremal codes. (ProQuest: ... denotes formulae/symbols omitted.)