Sharpening the linear programming bound for linear Lee codes

Abstract

Finding the largest code with a given minimum distance is one of the most basic problems in coding theory. A sharpening to the linear programming bound for linear codes in the Lee metric is introduced, which is based on an invariance-type property of Lee compositions of a linear code. Using this property, additional equality constraints are introduced into the linear programming problem, which give a tighter bound for linear Lee codes.

Semidefinite programming bounds for Lee codes

“…and x(S) = 0 else. Then x satisfies the conditions in (3), where the last condition is satisfied since…”

“…which is a 2 × 2-matrix. Then T ∅ M 2,∅ (z) ∅ = M 2,∅ (z) ∅,∅ = x(∅) = 1 by definition, see (3). Since v n = C ⊗n 1 is the all-ones vector, we have T ∅ M ∅ (z)v n = q n z ω 0 , where ω 0 ∈ Ω 2 is the (unique) D n q S n -orbit of a code of size 1.…”

“…Since there exists an H-invariant optimum solution, we can replace, for each ω ∈ Ω k and C ∈ ω, each variable x(C) by a variable z(ω). Hence, the matrices M k,D (x) become matrices M k,D (z) and we have considerably reduced the number of variables in (3).…”

“…The superscript b refers to a bound from Quistorff [20]. The superscript * refers to an upper bound matching the known lower bound: A L 7 (4, 5) = 49 is achieved by a linear code [3].…”