Let M denote the set Sn,q of n×n symmetric matrices with entries in Fq or the set Hn,qjavax.xml.bind.JAXBElement@d8d0cb2 of n×n Hermitian matrices whose elements are in Fqjavax.xml.bind.JAXBElement@3829cbf. Then M equipped with the rank distance dr is a metric space. We investigate d–codes in (M,dr) and construct d–codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an n–code of M, n even and n/2 odd, of size (3qn−qn/2)/2, and of a 2–code of size q6+q(q−1)(q4+q2+1)/2, for n=3. In the symmetric case, if n is odd, we provide better upper bound on the size of a 2–code. In the case when n=3 and q>2, a 2–code of size q4+q3+1 is exhibited. This provides the first infinite family of 2–codes of symmetric matrices whose size is larger than the largest possible additive 2–code and an answer to a question posed in [25, Section 7], see also [23, p. 176].

On symmetric and Hermitian rank distance codes

Pavese F.
2022

Abstract

Let M denote the set Sn,q of n×n symmetric matrices with entries in Fq or the set Hn,qjavax.xml.bind.JAXBElement@d8d0cb2 of n×n Hermitian matrices whose elements are in Fqjavax.xml.bind.JAXBElement@3829cbf. Then M equipped with the rank distance dr is a metric space. We investigate d–codes in (M,dr) and construct d–codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an n–code of M, n even and n/2 odd, of size (3qn−qn/2)/2, and of a 2–code of size q6+q(q−1)(q4+q2+1)/2, for n=3. In the symmetric case, if n is odd, we provide better upper bound on the size of a 2–code. In the case when n=3 and q>2, a 2–code of size q4+q3+1 is exhibited. This provides the first infinite family of 2–codes of symmetric matrices whose size is larger than the largest possible additive 2–code and an answer to a question posed in [25, Section 7], see also [23, p. 176].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/244750
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact