Subcodes with Prescribed Hull Dimensions in Linear Codes:Enumeration, Containment, and Asymptotic Distributions
Linhan Xiong, Hongwei Liu, Liren Lin
Mass formulas for prescribed hull dimensions have mainly been studied for linear codes viewed as subspaces of the whole space. We study internal enumeration for an arbitrary fixed parent code. Let $C$ be an $[n,k]_q$ linear code with an $\ell$-dimensional hull, and let $s$ be the subcode dimension. We determine the number of subcodes with prescribed dimension, hull dimension, and type over finite fields of both even and odd characteristic. We decompose the parent code into an LCD complement and its hull, and use projection to reduce enumeration to the corresponding problem inside the LCD complement. This yields explicit formulas refined by complete type and shows that the counts depend only on the relevant parameters, not on the particular parent code or its length. The known whole-space mass formulas and the enumeration of LCD subcodes of LCD codes arise as special cases. Motivated by code extension, we also study the dual containment problem: for a fixed code, we enumerate larger codes containing it with prescribed hull dimension and type. By duality, this problem is converted into the subcode enumeration problem already solved above, leading to corresponding conditional mass formulas. Finally, we investigate the asymptotic distributions arising from the exact enumeration formulas. When both the subcode dimension $s$ and $k-\ell-s$ tend to infinity, the hull distribution is governed by the LCD complement, and the limiting proportions are explicit constants determined by the field, the prescribed hull dimension, and the relevant type. When $s$ approaches or exceeds the dimension $k-\ell$ of the LCD complement, we determine the asymptotic proportion of subcodes attaining the minimum possible hull dimension and establish the associated concentration behavior. Analogous asymptotic results are obtained for the containment problem.
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