This monograph contributes to the existence theory of difference sets, cyclic irreducible codes and similar objects. Applications include the first substantial progress towards the Circulant Hadamard Matrix Conjecture and Ryser`s conjecture since decades.

1. Introduction: The nature of the problems.- The combinatorial structures in question.- Group rings, characters, Fourier analysis.- Number theoretic tools.- Algebraic-combinatorial tools. 2. The field descent: The fixing theorem.- Prescribed absolute value.- Bounding the absoute value.- The modulus equation and the class group. 3. Exponent bounds: Self-conjugacy exponent bounds.- Field descent exponent bounds. 4. Two-weight irreducible cyclic bounds: A necessary and sufficient condition.- All two-weight irreducible cyclic codes?- Partial proof of Conjecture 4.2.4.- Two-intersection sets and sub-difference sets

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