Giorgio Faina

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A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields. An LO code is nonshortening in the sense that one cannot remove any column from a parity-check matrix without increasing the code covering radius. Several q/sup m/-concatenating(More)
We give a geometric description of binary quantum stabilizer codes. In the case of distance d = 4 this leads to the notion of a quaternary quantum cap. We describe several recursive constructions for quantum caps, determine the quantum caps in PG(3, 4) and the cardinalities of quantum caps in PG(4, 4). ∗research partially supported NSA grant(More)
In this paper, the best parameters of quaternary additive codes of small length are determined using the geometric description. Only one open question remains for length les 13. Among the results obtained in this work are the nonexistence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]-code.
More than thirty new upper bounds on the smallest size t2(2, q) of a complete arc in the plane PG(2, q) are obtained for 169 ≤ q ≤ 839. New upper bounds on the smallest size t2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds(More)
A very difficult problem for complete caps in PG(r,q) is to determine their minimum size. The results on this topic are still scarce and in this paper we survey the best results now known. Furthermore, we construct new interesting sporadic examples of complete caps in PG(3, q) and in PG(4, q) such that their size are smaller than the currently known. As a(More)
New upper bounds on the smallest size t2(2, q) of a complete arc in the projective plane PG(2, q) are obtained for 853 ≤ q ≤ 5107 and q ∈ T1 ∪ T2, where T1 = {173, 181, 193, 229, 243, 257, 271, 277, 293, 343, 373, 409, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 529, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 661, 673, 677,(More)