389 KB23 KB200820252019Glasby, Stephen P.Praeger, Cheryl E.številka:1letnik:16str. 49-58ISSN:1855-3966COBISSID_HOST:18701657URN:URN:NBN:SI:doc-DQMW97BCenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneadimensiondimenzijadistancelinear codelinearna kodarazdaljaOn the parameters of intertwining codes|Let ?$F$? be a field and let ?$F^{r \times s}$? denote the space of ?$r \times s$? matrices over ?$F$?. Given equinumerous subsets ?$\mathcal A = \{A_i \mid i \in I\} \subseteq F^{r \times r}$? and ?$\mathcal B = \{B_i \mid i \in I\} \subseteq F^{s \times s}$? we call the subspace ?$C(\mathcal{A, B}) := \{X \in F^{r \times s} \mid A_iX = XB_i\ \mathrm{for}\ i \in I\}$? an intertwining code. We show that if ?$C(\mathcal{A, B}) \neq \{0\}$?, then for each ?$i \in I$?, the characteristic polynomials of ?$A_i$? and ?$B_i$? and share a nontrivial factor. We give an exact formula for ?$k = \dim(C(\mathcal{A, B}))$? and give upper and lower bounds. This generalizes previous work. Finally we construct intertwining codes with large minimum distance when the field is not "too small". We give examples of codes where ?$d = rs/k = 1/R$? is large where the minimum distance, dimension, and rate of the linear code ?$C(\mathcal{A, B})$? are denoted by ?$d, k,\ \mathrm{and}\ R = k/rs$?, respectivelyNaj bo? $F$? obseg, ?$F^{r \times s}$? pa prostor ?$r \times s$? matrik nad ?$F$?. Če sta podani enako močni podmnožici ?$\mathcal A = \{A_i \mid i \in I\} \subseteq F^{r \times r}$? in ?$\mathcal B = \{B_i \mid i \in I\} \subseteq F^{s \times s}$?, potem pravimo podprostoru? $C(\mathcal{A, B}) := \{X \in F^{r \times s} \mid A_iX = XB_i\ \mathrm{for}\ i \in I\}$? prepletna koda. Pokažemo, da če je ?$C(\mathcal{A, B}) \neq \{0\}$?, potem si, za vsak ?$i \in I$?, karakteristična polinoma matrik ?$A_i$? in ?$B_i$? delita netrivialen faktor. Izpeljemo natančno formulo za ?$k = \dim(C(\mathcal{A, B}))$? in podamo zgornjo in spodnjo mejo. To je posplošitev prejšnjih rezultatov. Konstruiramo prepletne kode z veliko minimalno razdaljo v primerih, ko obseg ni "premajhen" Podamo primere prepletov, kjer je ?$d = rs/k = 1/R$? velik, pri čemer so ?$d$?, ?$k$?, in ?$R = k/rs$? oznake za minimalno razdaljo, dimenzijo in velikost prepleta, v tem vrstnem reduTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije