Boundary of the binary mirror phenomenon and weakening the gap hypothesis

Characterize the exact boundary between binary linear codes that exhibit the mirror vanishing-band phenomenon and those that do not, and determine whether the condition that the weight-distribution coefficients satisfy A_{d+1}=\cdots=A_{d+t}=0 can be weakened or replaced by other local information.

Background

The paper proves that a binary linear [n,k,d] code with k\ge n-2d+1 and a local gap A_{d+1}=\cdots=A_{d+t}=0 must also have a mirror gap A_{2d+1}=\cdots=A_{2d+t}=0. It further shows that this phenomenon relies on the disjoint-support decomposition of non-minimal codewords, a property specific to binary codes, and that a direct q-ary analogue fails for MDS codes.

The unresolved issue is to identify precisely which binary codes satisfy this mirror phenomenon and which do not, and to determine whether the strong consecutive-gap assumption immediately above the minimum distance can be relaxed or replaced with different local information about the weight distribution.