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We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl’s database [13].

- Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
- IEEE Transactions on Information Theory
- 2005

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)

- Jürgen Bierbrauer, Daniele Bartoli, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco, Yves Edel
- Des. Codes Cryptography
- 2014

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)

- Jürgen Bierbrauer, Yves Edel, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
- IEEE Transactions on Information Theory
- 2009

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.

- Alexander A. Davydov, Giorgio Faina, +5 authors Fernanda Pambianco
- 2009

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)

- Giorgio Faina, Stefano Marcugini, Alfredo Milani, Fernanda Pambianco
- Ars Comb.
- 1998

- Giorgio Faina, Massimo Giulietti
- Discrete Mathematics
- 2003

In the Galois projective plane of square order q, we show the existence of small dense (k; 4)-arcs whose points lie on two conics for q odd, and on two hyperovals for q even. We provide an explicit construction of (4 √ q− 4; 2)-arcs for q even, and we also show that they are complete as far as q6 1024. c © 2002 Elsevier Science B.V. All rights reserved.

- Giorgio Faina, Fernanda Pambianco
- Discrete Mathematics
- 1997

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)

- Daniele Bartoli, Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
- Discrete Mathematics
- 2012

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)

- Fernanda Pambianco, Daniele Bartoli, Giorgio Faina, Stefano Marcugini
- Electronic Notes in Discrete Mathematics
- 2013