OPTIMALITY OF CODES BASED ON CROSSED PRODUCT ALGEBRAS GREGORY BERHUY, RICHARD SLESSOR Abstract. In this paper, we explain how to construct reliable codes for wireless communication channels using crossed product division algebras, and we prove the optimality of the codes already constructed on cyclic algebras and biquadratic crossed products. Contents Introduction 1 1. From codes to crossed product algebras 3 1.1. Modelling a communication channel 3 1.2. Algebra based codes. 7 1.3. Codes based on crossed product K-algebras. 11 2. Ideal lattices 15 2.1. Generalities on hermitian lattices 15 2.2. Complex ideal lattices 17 2.3. Minimum determinant of a crossed product based code 22 3. Optimality of codes based on cyclic K-algebras 25 3.1. Preliminaries 25 3.2. The case n = 4 27 3.3. The case n = 6 31 4. Optimality of codes based on biquadratic crossed products 35 References 44 Introduction Within the last few years we have seen a notable increase in the use of wireless communication, which has led to the need for higher data rates. In view of this multiple antenna communication systems have 1
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