Extended Cyclic Codes Sandwiched Between Reed-Muller Codes
Yan Xu, Changjiang Ji, Ran Tao, Sihuang Hu
TL;DR
This paper extends the construction of universally strongly perfect lattices by applying a specialized Construction D to chains of extended cyclic codes between Reed-Muller codes, and explicitly determines minimum vectors for certain cases.
Contribution
It generalizes previous lattice constructions to generalized Reed-Muller codes and explicitly finds minimum vectors in specific instances.
Findings
Construction of new strongly perfect lattices between Barnes-Wall lattices.
Explicit determination of minimum vectors for certain generalized Reed-Muller code cases.
Extension of previous cyclic code-based lattice constructions.
Abstract
The famous Barnes-Wall lattices can be obtained by applying Construction D to a chain of Reed-Muller codes. By applying Construction to a chain of extended cyclic codes sandwiched between Reed-Muller codes, Hu and Nebe (J. London Math. Soc. (2) 101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes-Wall lattices. In this paper, we first extend their construction to generalized Reed-Muller codes, and then explicitly determine the minimum vectors of those new sandwiched Reed-Muller codes for some special cases.
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