Introduction Motivation Our contributions Cyclic Subspace Codes Via Subspace Polynomials Kamil Otal and Ferruh Özbudak Middle East Technical University Design and Application of Random Network Codes (DARNEC’15) November 4-6, 2015 / Istanbul, Turkey. 1 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Motivation Our contributions Outline Introduction 1 Subspace codes Cyclic subspace codes Subspace Polynomials Motivation 2 Literature Related work Our goal 3 Our contributions A generalization: More codewords One more generalization: More diverse parameters 2 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Subspace codes Consider the following notations and definitions. q : a prime power, q q F q : the finite field of size q , F q F q N , k : positive integers such that 1 < k < N , N , k N , k P q ( N ) P q ( N ) P q ( N ) : the set of all subspaces of F N q , G q ( N , k ) G q ( N , k ) G q ( N , k ) : the set of k -dimensional subspaces in P q ( N ) , Subspace distance : d ( U , V ) ∶= dim U + dim V − 2dim ( U ∩ V ) for all U , V ∈ P q ( N ) . 3 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Subspace codes Subspace code : A nonempty subset C of P q ( N ) with the subspace distance. Constant dimension code : A subspace code C if C ⊆ G q ( N , k ) . Distance of a code : d (C) ∶= min { d ( U , V ) ∶ U , V ∈ C and U ≠ V } . 4 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Cyclic subspace codes Consider F q N instead of F N q equivalently (and richly). F ∗ F ∗ F ∗ q N : the set of nonzero elements of F q N . q N q N Cyclic shift of a codeword U by α ∈ F ∗ q N : α U ∶= { α u ∶ u ∈ U } . It is easy to show that the cyclic shift is also a subspace of the same dimension. Orbit of a codeword U : Orb ( U ) ∶= { α U ∶ α ∈ F ∗ q N } . It is easy to show that orbits form an equivalence relation in G q ( N , k ) and so in P q ( N ) . Cyclic (subspace) code : A subspace code C if Orb ( U ) ⊆ C for all U ∈ C . 5 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Cyclic subspace codes The following theorem is well known. Theorem Let U ∈ G q ( N , k ) . F q d is the largest field such that U is also F q d -linear (i.e. linear over F q d ) if and only if ∣ Orb ( U )∣ = q N − 1 q d − 1 . 6 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Cyclic subspace codes Let d denote the largest integer where U is also F q d -linear. Full length orbit : An orbit if d = 1. Degenerate orbit : An orbit which is not full length. Remark that d divides both N and k . More explicitly, U ∈ G q ( N , k ) U ∈ G q d ( N / d , k / d ) . ⇔ Therefore, it is enough to study on full length orbits. 7 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Subspace Polynomials Linearized polynomial ( q -polynomial) : F ( x ) = α s x q s + α s − 1 x q s − 1 + ... + α 0 x ∈ F q N [ x ] for some nonnegative integer s . The roots of F form a subspace of an extension of F q N . The multiplicity of each root of F is the same, and equal to q r for some nonnegative integer r ≤ s . Explicitly, r is the smallest integer satisfying α r is nonzero. 8 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Subspace codes Motivation Cyclic subspace codes Our contributions Subspace Polynomials Subspace Polynomials Subspace polynomial : A monic linearized polynomial such that splits completely over F q N , has no multiple root (equivalently α 0 ≠ 0). More explicitly, it is the polynomial ( x − u ) ∏ u ∈ U where U is a subspace of F q N . 9 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Literature Motivation Related work Our contributions Our goal Literature Subspace codes, particularly constant dimension codes, have been intensely studied in the last decade due to their application in random network coding 1 . Cyclic subspace codes are useful in this manner due to their efficient encoding and decoding algorithms. Some recent studies about cyclic codes and their efficiency are: –> A. Kohnert and S. Kurz; Construction of large constant dimension codes with a prescribed minimum distance , Lecture Notes Computer Science, vol. 5395, pp. 31–42, 2008. –> T. Etzion and A. Vardy; Error correcting codes in projective space , IEEE Trans. on Inf. Theory, vol. 57, pp. 1165–1173, 2011. 1 R. Kötter and F. R. Kschischang; Coding for errors and erasures in random network coding , IEEE Trans. on Inf. Theory, vol. 54, pp. 3579–3591, 2008. 10 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Literature Motivation Related work Our contributions Our goal Literature –> A.-L. Trautmann, F . Manganiello, M. Braun and J. Rosenthal; Cyclic orbit codes , IEEE Trans. on Inf. Theory, vol. 59, pp. 7386–7404, 2013. –> M. Braun, T. Etzion, P . Ostergard, A. Vardy and A. Wasserman; Existence of q-analogues of Steiner systems , arXiv:1304.1462, 2013. –> H. Gluesing-Luerssen, K. Morrison and C. Troha; Cyclic orbit codes and stabilizer subfields , Adv. in Math. of Communications, vol. 25, pp. 177–197, 2015. –> E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv; Subspace polynomials and cyclic subspace codes ; arXiv:1404.7739v3, 2015. (Also in ISIT 2015, pp. 586-590.) 11 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Literature Motivation Related work Our contributions Our goal Related work Theorem 1 a a E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv; Subspace polynomials and cyclic subspace codes ; arXiv:1404.7739v3, 2015. (Also in ISIT 2015, pp. 586-590.) Let n be a prime, γ be a primitive element of F q n , F q N be the splitting field of the polynomial x q k + γ q x q + γ x , U ∈ G q ( N , k ) is this polynomial’s kernel. 12 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Literature Motivation Related work Our contributions Our goal Related work Theorem 1 (cont’d.) Then { α U q i ∶ α ∈ F ∗ n − 1 C ∶= q N } ⋃ i = 0 q − 1 and minimum distance 2 k − 2. is a cyclic code of size n q N − 1 13 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction Literature Motivation Related work Our contributions Our goal Our goal Our goal is to generalize their result in two directions: Can we insert more orbits (i.e. more codewords)? Can we use other types of subspace polynomials (and hence cover more diverse values of length N )? 14 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction A generalization: More codewords Motivation One more generalization: More diverse parameters Our contributions A generalization: More codewords Theorem 2 Let n and r be positive integers such that r ≤ q n − 1 and let - γ 1 ,...,γ r be distinct elements of F ∗ q n , - T i ( x ) ∶= x q k + γ q i x q + γ i x for all i ∈ { 1 ,..., r } , - N i be the degree of the splitting field of T i for all i ∈ { 1 ,..., r } , - U i ⊆ F q Ni be the kernel of T i for all i ∈ { 1 ,..., r } , - N be the least common multiple of N 1 ,..., N r . 15 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction A generalization: More codewords Motivation One more generalization: More diverse parameters Our contributions A generalization: More codewords Theorem 2 (cont’d.) Then the code C ⊆ G q ( N , k ) given by C = r { α U i ∶ α ∈ F ∗ q N } ⋃ i = 1 q − 1 and the minimum distance 2 k − 2. is a cyclic code of size r q N − 1 Moreover, if γ i and γ j are conjugate as γ i = γ q m for some integer j m , then N i = N j and U i = U q m . j 16 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
Introduction A generalization: More codewords Motivation One more generalization: More diverse parameters Our contributions A generalization: More codewords Corollary 1 Let n be a positive integer and γ 1 = γ,γ 2 = γ q ,...,γ n = γ q n − 1 ∈ F q n for some irreducible element γ of F q n . Then, by using the construction in Theorem 2, we can produce a cyclic code of size nq N − 1 q − 1 and the minimum distance 2 k − 2. Resulting code is the same with the one in Theorem 1. 17 / 26 K. Otal and F. Özbudak Cyclic Subspace Codes Via Subspace Polynomials
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