We consider the extendability of linear codes over the field of order 4 whose codewords have at most four weights modulo 16. For instance, we prove that every n,k,d4 code with d≡−3(mod16), k≥3, is ...extendable if all possible weights of codewords are congruent to 0,−1,−2,−3(mod16). Such a code with d≡−1 or −2(mod16) is extendable as well except for some cases.
A constant-dimension code (CDC) is a set of subspaces of constant dimension in a common vector space with upper bounded pairwise intersection. We improve and generalize two constructions for CDCs, ...the improved linkage construction and the parallel linkage construction , to the generalized linkage construction and the multiblock generalized linkage construction which in turn yield many improved lower bounds for the cardinalities of CDCs; a quantity not known in general.
We prove upper bounds for the cardinality of constant dimension codes (CDC) which contain a lifted maximum rank distance (LMRD) code as a subset. Thereby we cover all parameters fulfilling ...<inline-formula> <tex-math notation="LaTeX">k\lt 3d/2 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> is the codeword dimension and <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is the minimum subspace distance. The proofs of the bounds additionally show that an LMRD code <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> can be unioned with a CDC <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> (of fitting parameters) without violating the subspace distance condition iff each codeword of <inline-formula> <tex-math notation="LaTeX">C </tex-math></inline-formula> intersects the special subspace of <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> in at least dimension <inline-formula> <tex-math notation="LaTeX">d/2 </tex-math></inline-formula>. This connection is used to find the new largest and sometimes bound achieving CDCs for small parameters.
We consider the extendability of linear codes over F4, the field of order four. Let C be n,k,d4 code with d≡1(mod4), k≥3. The weight spectrum modulo 4 (4-WS) of C is defined as the ordered 4-tuple ...(w0,w1,w2,w3) with w0=13∑4|i>0Ai, wj=13∑i≡j(mod4)Ai for j=1,2,3. We prove that C is 3-extendable if w0+w2=θk−2 and if either (a) w1−w0<4k−2+4−θk−3; (b) w1−w0>10⋅4k−3−θk−3 or (c) (w0,w1)=(θk−3,6⋅4k−3). We also give a sufficient condition for the l-extendability of n,k,d4 codes with d≡4−l(mod4), k≥3 for l=1,2,3 when w0+w2=θk−2+2⋅4k−2.
The packings of PG(3,3) Betten, Anton
Designs, codes, and cryptography,
06/2016, Letnik:
79, Številka:
3
Journal Article
Recenzirano
Packings of
PG
(
3
,
q
)
are closely related to Kirkman’s problem of the 15 schoolgirls from 1850 and its generalizations:
Fifteen young ladies in a school walk out three abreast for seven days in ...succession: it is required to arrange them daily so that no two shall walk twice abreast.
The packings of
PG
(
3
,
2
)
give rise to two of seven solutions of Kirkman’s problem. Here, we continue the problem of classifying packings of
PG
(
3
,
q
)
by settling the case
q
=
3
.
We find that there are exactly 73,343 packings of
PG
(
3
,
3
)
.
An n,k,dq code C is l-extendable if C can be extended to an n+l,k,d+lq code. We give some new sufficient conditions for the l-extendability of n,k,d4 codes with d≢0(mod4) using the known results ...about odd sets in PG(k−1,4). The 3-extendability of quaternary linear codes is investigated through their diversities for the first time.
A set
K
in PG(
r
, 4),
r
≥ 2, is
odd
if every line meets
K
in an odd number of points. An odd set
K
in PG(
r
, 4),
r
≥ 3, is
FH-free
if there is no plane meeting
K
in a Fano plane or in a ...non-singular Hermitian curve. We prove that an odd set
K
contains a hyperplane of PG(
r
, 4) if and only if
K
is FH-free. As an application to coding theory, a new extension theorem for quaternary linear codes is given.
PDF
