IV code. If the case $\delta(C) < s(C)$ occurs, then $C$ is an extremal Type III $[764,382,192]$ code which has $\delta(C)=3$ and $s(C)=4,5$ or $6$.

This paper is organized as follows. In Section \[sec:pre\], we give definitions and some basic properties of self-dual codes and $t$-designs and review the concept of harmonic weight enumerators and some lemmas which are used in the proof of the main results. In Section \[sec: proof of thm1.3\] and \[sec: proof of thm1.4\], we give proofs of Theorem \[thm:main upper bound III\] and \[thm:main upper bound IV\], respectively.

All computer calculations in this paper were done with the help of Mathematica [@Mathematica].

Preliminaries {#sec:pre}
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Codes and the support $t$-designs
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Let ${{\mathbb F}}_q$ be the finite field of $q$ elements. A linear code $C$ of length $n$ is a linear subspace of ${{\mathbb F}}_{q}^{n}$. For $q=3$, an inner product $({x},{y})$ on ${{\mathbb F}}_q^n$ is given by $$(x,y)=\sum_{i=1}^nx_iy_i,$$ where $x,y\in {{\mathbb F}}_q^n$ with $x=(x_1,x_2,\ldots, x_n)$ and $y=(y_1,y_2,\ldots, y_n)$. The Hermitian inner product $({x},{y})$ on ${{\mathbb F}}_4^n$ is given by $$(x,y)_H=\sum_{i=1}^nx_iy_i^2,$$ where $x,y\in {{\mathbb F}}_4^n$ with $x=(x_1,x_2,\ldots, x_n)$ and $y=(y_1,y_2,\ldots, y_n)$. The dual of a linear code $C$ is defined as follows: for $q=3$, $$C^{\perp}=\{{y}\in {{\mathbb 