A characterization of some \([n,k,d;q]\)-codes meeting the Griesmer bound using a minihyper in a finite projective geometry

DOI10.1016/0012-365X(93)90404-HzbMath0770.94006MaRDI QIDQ1802134

Noboru Hamada

Publication date: 13 September 1993

Published in: Discrete Mathematics (Search for Journal in Brave)

zbMATH Keywords

hyperplanesGriesmer boundfinite projective geometryminihyper

Mathematics Subject Classification ID

Linear codes (general theory) (94B05) Bounds on codes (94B65) Combinatorial structures in finite projective spaces (51E20)

Related Items (53)

A study of \((x(q + 1), x; 2, q)\)-minihypers ⋮ On the minimum length of some linear codes ⋮ A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound ⋮ A characterization of some \(\{3v_{\mu+ 1}, 3v_ \mu; k-1, q\}\)-minihypers and some \([n, k, q^{k-1}- 3q^ \mu; q\)-codes \((k\geq 3\), \(q\geq 5\), \(1\leq \mu< k-1)\) meeting the Griesmer bound] ⋮ Uniqueness of \([87,5,57; 3\)-codes and the nonexistence of \([258,6,171; 3]\)-codes] ⋮ A characterization of some \(\{ 3v_ 2+v_ 3,3v_ 1+v_ 2; 3,3\}\)-minihypers and some \([15,4,9; 3\)-codes with \(B_ 2=0\)] ⋮ A characterization of some \(\{ 3v_ 1+v_ 3,3v_ 0+v_ 2; 3,3\}\)-minihypers and its applications to error-correcting codes ⋮ The nonexistence of \([71,5,46;3\)-codes] ⋮ Unnamed Item ⋮ Nonexistence of some ternary linear codes with minimum weight \(-2\) modulo 9 ⋮ Extension theorems for linear codes over finite fields ⋮ New upper bounds and constructions of multi-erasure locally recoverable codes ⋮ On the minimum length of ternary linear codes ⋮ On codes meeting the Griesmer bound. ⋮ Unnamed Item ⋮ A characterization of \(\{ 2\upsilon{}_{\alpha{}+1}+2\upsilon{}_{\beta{}+1},2\upsilon_ \alpha{}+2\upsilon{}_ \beta{} ;t,q\}\)-minihypers in PG\((t,q)(t\geq 2,q\geq 5\) and \(0\leq\alpha{}<\beta{}<t)\) and its applications to error- correcting codes ⋮ New sufficient conditions for the extendability of quaternary linear codes ⋮ Parameters for which the Griesmer bound is not sharp ⋮ Binary and ternary LCD codes from projective spaces ⋮ On optimal non-projective ternary linear codes ⋮ A characterization of some \(\{2v_{\alpha{}+1}+v_{\gamma{}+1},2v_ \alpha{}+v_ \gamma{};k-1,3\}\)-minihypers and some \((n,k,3^{k-1}- 2\cdot{}3^ \alpha{}-3^ \gamma{};3)\)-codes \((k\geq{}3,\;0 \leq{}\alpha{}< \gamma{}< k-1)\) meeting the Griesmer bound ⋮ A new extension theorem for ternary linear codes and its application ⋮ On the minimum length of linear codes over the field of 9 elements ⋮ Projective codes meeting the Griesmer bound ⋮ The nonexistence of ternary [97, 6, 63 codes] ⋮ On the minimum size of some minihypers and related linear codes ⋮ Optimal fractional factorial plans using minihypers ⋮ A classification result on weighted \(\{\delta v_{\mu +1},\delta v_{\mu};N,p^{3}\}\)-minihypers ⋮ On the minimum length of some linear codes of dimension 5 ⋮ The non-existence of Griesmer codes with parameters close to codes of Belov type ⋮ A new extension theorem for 3-weight modulo \(q\) linear codes over \({\mathbb{F}_q}\) ⋮ On weighted minihypers in finite projective spaces of square order ⋮ Characterization results on arbitrary non-weighted minihypers and on linear codes meeting the Griesmer bound ⋮ Tight sets, weighted \(m\)-covers, weighted \(m\)-ovoids, and minihypers ⋮ A class of optimal linear codes of length one above the Griesmer bound ⋮ Extendability of 3-weight (mod \(q\)) linear codes over \(\mathbb F_q\) ⋮ A survey of recent works with respect to a characterization of an (n,k,d;q)-code meeting the Griesmer bound using a min\(\cdot hyper\) in a finite projective geometry ⋮ On optimal linear codes of dimension 4 ⋮ On optimal linear codes over \(\mathbb F_5\) ⋮ Arcs, blocking sets, and minihypers ⋮ Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes ⋮ Nonexistence of linear codes meeting the Griesmer bound ⋮ A new extension theorem for linear codes ⋮ On the nonexistence of some quaternary linear codes meeting the Griesmer bound ⋮ The nonexistence of ternary [79, 6, 51 codes] ⋮ The packing problem in statistics, coding theory and finite projective spaces ⋮ Nonexistence of \([n,5,d_q\) codes attaining the Griesmer bound for \(q^4-2q^2-2q+1\leq d\leq q^4-2q^2-q\).] ⋮ On the minimum length of quaternary linear codes of dimension five ⋮ A new class of nonbinary codes meeting the Griesmer bound ⋮ A characterization of some minihypers in a finite projective geometry PG(t,4) ⋮ On a particular class of minihypers and its applications. III: Applications ⋮ On a particular class of minihypers and its applications. II: Improvements for \(q\) square ⋮ Minimal covers of \(Q^+(2n+1,q)\) by \((n-1)\)-dimensional subspaces

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