ZETA POLYNOMIALS OF TYPE IV CODES OVER RINGS OF ORDER FOUR

Abstract

We extend the definition of zeta function and zeta polynomial to codes definedover finite rings with respect to a specified weight function. Moreover, we also investigate the Riemann hypothesis analogue for Type IV codes over any of the rings Z₄, F₂ + uF₂ and F₂ + νF₂. Although, for small lengths, there are only a few actual Type IV codes over Z₄, F₂ + uF₂ or F₂ + νF₂ that satisfy the Hamming distance upper bound 2(1 + n/6), we will show that zeta polynomials corresponding to these weight enumerators that meet this bound satisfy the Riemann hypothesis analogue property.