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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})] is determined. As an application, the weight distribution of the binary cyclic code $cC$, with parity-check polynomial $h_1(x)h_2(x)h_3(x)$ where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $pi^{-1}$, $pi^{-(2^k+1)}$ and $pi^{-(2^{2k}+1)}$ respectively for a primitive element $pi$ of $bF_q$, is also determined.
In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $Tr(x^d)$ over $F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $gcd(d, 2^{m}-1)=1$.
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 dist
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
One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the b
In this paper, we apply two-to-one functions over $mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $left(x^{2^t}+xright)^e$ with $gcd(t,