Qadratic residue codes over 𝔽p + u𝔽p + v𝔽p∗

Abstract

In this paper, we study quadratic residue (QR) codes of prime length [Formula: see text] over the ring [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length [Formula: see text] over [Formula: see text], we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good [Formula: see text]-ary codes as Gray images of QR codes over [Formula: see text] by considering the case when [Formula: see text] is an odd prime.