Decreasing Monomial Codes over the Quaternary Ring ℤ4

Abstract This paper extends the work on monomial codes over 𝔽 2 Bardet, et al .[1] to the ring ℤ 4 , developing an algebraic framework for constructing and analyzing quaternary monomial codes. In particular, the decreasing monomial codes in R m , the quotient ring over the polynomial ring ℤ 4 with indeterminates x 0 , …, x m –1 modulo the ideal ( x 2 0 — x 0 , …, x 2 m –1 – x m –1 ), defined via evaluations ordered by a refined partial order on monomials to determine minimal generating sets. A key result is the characterization of quaternary Reed-Muller codes RM( r,m ) as decreasing monomial codes, generated by monomials selected according to degree and coefficient constraints. This construction enables systematic code generation, and examples computed in MAGMA confirm that several of the resulting codes meet known upper bounds.