The Geometry of Two-Weight Codes Over ℤ<sub><i>p</i></sub><sup><i>m</i></sup>
Minjia Shi, Thomas Honold, Patrick Solé, Yunzhen Qiu, Rongsheng Wu, Zahra Sepasdar
Abstract
We investigate fat projective linear codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb Z}_{p^{m}}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geqslant 2$ </tex-math></inline-formula> , with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$ </tex-math></inline-formula> . This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry.