On Construction of Linear (Euclidean) Hull Codes over Finite Extensions Binary Fields
Sanjit Bhowmick, Deepak Kumar Dalai, Sihem Mesnager
TL;DR
This work studies the hull of a linear code $\mathcal{C}=[n,k,d]$ over a finite field with emphasis on constructing hulls of dimension $\ell+1$ from an existing $\ell$-dimensional hull, and in particular extends results over extended binary fields ${\rm I\!F}_{2^t}$ to obtain 1-dimensional hulls via weak conditions. Building on Chen's 2023 findings that LCD codes with $d\ge2$ are equivalent to 1-dimensional hull codes under a weak condition, the authors show analogous hull-lifting for general $\ell$ and provide explicit generator-timeform constructions, such as $G=\left(10P_1a0I_{k-1}P_2b\right)$, and coordinate-scaling methods to realize a 1D hull. They further develop a general method to increase hull dimension by 1 using a parameterized family $\mathcal{C}_{\lambda}$ and derived $G_{\lambda},H_{\lambda}$, with determinant considerations ensuring a nontrivial intersection with the dual. The results yield a toolkit for designing hull-controlled codes and open avenues for hull-based cryptographic and coding applications, extending the concept of hull variation beyond 1D to higher dimensions under verifiable weak conditions.
Abstract
The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to check the permutation equivalence of two linear codes and compute a linear code's automorphism group. Research has shown that these algorithms are very effective when the hull size is small. Linear complementary dual (LCD) codes have the smallest hulls, while codes with a one-dimensional hull have the second smallest. A recent notable paper that directs our investigation is authored by H. Chen, titled ``On the Hull-Variation Problem of Equivalent Linear Codes", published in IEEE Transactions on Information Theory, volume 69, issue 5, in 2023. In this paper, we first explore the one-dimensional hull of a linear code over finite fields. Additionally, we demonstrate that any LCD code over an extended binary field \( \FF_q \) (where \( q > 3 \)) with a minimum distance of at least $2$ is equivalent to the one-dimensional hull of a linear code under a specific weak condition. Furthermore, we provide a construction for creating hulls with \( \ell + 1 \)-dimensionality from an \( \ell \)-dimensional hull of a linear code, again under a weak condition. This corresponds to a particularly challenging direction, as creating \( \ell \)-dimensional hulls from \( \ell + 1 \)-dimensional hulls. Finally, we derive several constructions for the \( \ell \)-dimensional hulls of linear codes as a consequence of our results.