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The Doob scheme $D(m,n+n)$ is a metric association scheme defined on $E_4^m times F_4^{n}times Z_4^{n}$, where $E_4=GR(4^2)$ or, alternatively, on $Z_4^{2m} times Z_2^{2n} times Z_4^{n}$. We prove the MacWilliams identities connecting the weight distributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code $C$ in the Doob scheme $D(m,n+n)$ is related by the MacWilliams identities with the weight distribution of the dual code $C^perp$ in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m times F_4^{n}$, the weight distributions of $C$ and $C^perp$ in the same scheme are also connected.
The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m times (Z_2^2)^{n} ti
Let $q=2^n$, $0leq kleq n-1$, $n/gcd(n,k)$ be odd and $k eq n/3, 2n/3$. In this paper the value distribution of following exponential sums [sumlimits_{xin bF_q}(-1)^{mathrm{Tr}_1^n(alpha x^{2^{2k}+1}+beta x^{2^k+1}+ga x)}quad(alpha,beta,gain bF_{q})]
For any integer $rho geq 1$ and for any prime power q, the explicit construction of a infinite family of completely regular (and completely transitive) q-ary codes with d=3 and with covering radius $rho$ is given. The intersection array is also compu
Suppose that $mathcal{P}$ is a property that may be satisfied by a random code $C subset Sigma^n$. For example, for some $p in (0,1)$, $mathcal{P}$ might be the property that there exist three elements of $C$ that lie in some Hamming ball of radius $
A code ${cal C}$ is $Z_2Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectivel