Weight hierarchies of linear codes satisfying the almost chain condition
Tóm tắt
The weight hierarchy of a linear [n, k; q] codeC overGF(q) is the sequence (d
1,d
2, …,d
k) whered
r is the size of the smallest support of anr-dimensional subcode ofC. An [n, k; q] code satisfies the chain condition if there exists subcodesD
1⊂D
2⊂…⊂D
k=C ofC such thatD
r has dimensionr and support of sized
r for allr. Further,C satisfies the almost chain condition if it does not satisfy the chain condition, but there exist subcodesD
r of dimensionr and support of sized
r for allr such thatD
2⊂D
3⊂…⊂D
k=C andD
1⊂D
3. A simple necessary condition for a sequence to be the weight hierarchy of a code satisfying the almost chain condition is given. Further, explicit constructions of such codes are given, showing that in almost all cases, the necessary conditions are also sufficient.
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