Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (derived through a linear programming relaxation approach) which states that the rate $R(\delta)$ corresponding to a relative distance of $\delta$ is such that: \begin{equation*}R(\delta) \leq H_2(\frac{1}{2}-\sqrt{\delta(1-\delta)}) \end{equation*} where H is the (binary) entropy function. (A slight improvement of the above exists in the binary case, but within the same framework) In the case of q-ary codes, i.e. codes over $\mathbb F _q ^n$, the above bound is generalized to: \begin{equation*}R(\delta) \leq H_q(\frac{1}{q}(q-1-(q-2)\delta-2\sqrt{(q-1)\delta(1-\delta)})) \end{equation*} My question is as follows: For larger alphabet size q, the above bound seems to weaken significantly. In fact, observing the growth of the above bound as $q \rightarrow \infty$, we see that: \begin{equation*} R(\delta) \leq 1-\delta+\mathcal{O}(\frac{1}{\log{q}}) \end{equation*} Thus, it seems to get worse than even the Singleton bound $R(\delta) \leq 1-\delta$.

So which is the best bound for large alphabet size $q$? Or am I wrong in the above conclusion that the MRRW bound is worse than even the Singleton bound for larger $q$? Also, could someone direct me to references for comparisons of different upper bounds for larger $q$? I am able to find reliable comparisons only for $q=2$.


1 Answer 1


This is not the best bound even for $q=2$; in fact, this is not the best bound derived from the Delsarte linear program; see the paper "On the optimum of Delsarte's linear program" by Samorodnitsky (1998).

Thus, a better analysis of the linear program is likely to improve the bounds over larger $q$. Even for $q=2$, this is a complicated analysis, so I don't know whether anybody has performed it for larger $q$.


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