Additive Cyclic Codes Funda Ozdemir Faculty of Engineering and - PowerPoint PPT Presentation

Additive Cyclic Codes Funda ¨ Ozdemir Faculty of Engineering and Natural Sciences Sabancı University, Istanbul SP Coding School, January 19-30, 2015 Funda ¨ Ozdemir Additive Cyclic Codes

Introduction J.Bierbrauer generalized the theory of cyclic codes from the category of linear codes to the category of additive codes in 2002. We will state Bierbrauer’s BCH bound on the minimum distance of additive cyclic codes. Our goal is to improve the bound on the minimum distance of these codes. Funda ¨ Ozdemir Additive Cyclic Codes

Setting Consider F q with q = p e . Let n | q r − 1 and W = < α > be a multiplicative subgroup of F ∗ q r . Fix A = { i 1 , ..., i s } ⊆ Z / n Z . Define the F q -linear space of polynomials P ( A ) := { a 1 x i 1 + · · · + a s x i s : a 1 , ..., a s ∈ F q r } Set B ( A ) := { ( f ( α 0 ) , ..., f ( α n − 1 )) : f ( x ) ∈ P ( A ) } Define a surjective F q -linear mapping F m φ : F q r → q x �→ ( Tr ( γ 1 x ) , ..., Tr ( γ m x )) for some subset { γ 1 , ..., γ m } ⊂ F q r , where Tr denotes the trace function from F q r to F q . The set { γ 1 , ..., γ m } is linearly independent over F q since φ is onto. q ) n in the usual way. Extend φ to a mapping : F n q r → ( F m Funda ¨ Ozdemir Additive Cyclic Codes

Additive Cyclic Codes Definition Define the additive cyclic code over the alphabet F m q with length n as ( φ ( B ( A ))) ⊥ C ( A ) : = { ( φ ( f ( α 0 )) , ..., φ ( f ( α n − 1 ))) : f ( x ) ∈ P ( A ) } ⊥ ⊆ ( F m q ) n = { ( Tr ( γ 1 f ( α 0 )) , ..., Tr ( γ m f ( α 0 )); ... ; Tr ( γ 1 f ( α n − 1 )) , ..., Tr ( γ m f ( α n − 1 ))) : f ( x ) ∈ P ( A ) } ⊥ = F mn ⊆ q The code C ( A ) is not linear over its alphabet F m q . If we view C in F mn as above, then it is q F q -linear. C ( A ) is cyclic : Since the dual code of a cyclic code is also cyclic, it is enough to show that C ( A ) ⊥ is cyclic. Consider the codeword c f = ( φ ( f ( α 0 )) , ..., φ ( f ( α n − 1 ))) s λ j x i j ∈ P ( A ). determined by f ( x ) = � j =1 s λ j α − i j x i j ∈ P ( A ), we have: For g ( x ) = � j =1 ( φ ( f ( α n − 1 )) , φ ( f ( α 0 )) , ..., φ ( f ( α n − 2 ))) = ( φ ( g ( α 0 )) , φ ( g ( α )) , ..., φ ( g ( α n − 1 ))) Linear cyclic codes correspond to the special case when m = 1 and φ ( x ) = Tr ( x ). Funda ¨ Ozdemir Additive Cyclic Codes

BCH Bound Definition A ⊆ Z / n Z is an interval if there is a generator (an integer j , coprime with n ) of Z / n Z such that A = { jl , j ( l + 1) , ..., j ( l + i − 1) } , for some l (mod n ). In the special case A = { i , i + 1 , ..., j } , the short notation A = [ i , j ] is used. Theorem (Bierbrauer’s BCH bound) If A contains an interval of size t (mod n), then the minimum distance of C ( A ) is ≥ t + 1 . Goal: Improve the minimum distance bound for additive cyclic codes! Funda ¨ Ozdemir Additive Cyclic Codes