This paper proposes a new class of signature codes for synchronous optical code-division multiple access (CDMA) and describes a general method for construction of the codes. The proposed codes can be obtained from generalized modified prime sequence codes (GMPSCs) based on extension fields GF(q), where q=p^m, p is a prime number, and m is a positive integer. It has been reported that optical CDMA systems using GMPSCs remove not only multi-user interference but also optical interference (e.g., background light) with a constant intensity during a slot of length q². Recently, the authors have reported that optical CDMA systems using GMPSCs also remove optical interference with intensity varying by blocks with a length of q. The proposed codes, referred to as p-chip codes in general and chip-pair codes in particular for the case of p=2, have the property of removing interference light with an intensity varying by shorter blocks with a length of p without requiring additional equipment. The present paper also investigates the algebraic properties and applications of the proposed codes.