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Abstract for Mihai Caragiu,
On a Class of Constant Weight Codes
For any odd prime power $q$ we first construct a certain
non-linear binary code $C(q,2)$ having $(q^2-q)/2$ codewords
of length $q$ and weight $(q-1)/2$ each, for which the Hamming
distance between any two distinct codewords is in the range
$[q/2-3\sqrt q/2,\ q/2+3\sqrt q/2]$ that is, `almost constant'.
Moreover, we prove that $C(q,2)$ is distance-invariant. Several
variations and improvements on this theme are then pursued.
Thus, we produce other classes of binary codes $C(q,n)$, $n\geq 3$,
of length $q$ that have `almost constant' weights and distances,
and which, for fixed $n$ and big $q$, have asymptotically $q^n/n$
codewords. Then we prove the possibility of extending our codes by
adding the complements of their codewords. Also, by using results
on Artin $L-$series, it is shown that the distribution ofthe $0$'s
and $1$'s in the codewords we constructed is quasi-random.
Our construction uses character sums associated with the quadratic
character $\chi$ of $F_{q^n}$ in which the range of summation
is $F_q$. Relations with the duals of the double error
correcting BCH codes and the duals of the Melas codes are also discussed.
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