Introduction to Space-Time 2017/Talks/   Construction A Non-associative Algebras Introduction

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Text of Introduction to Space-Time 2017/Talks/   Construction A Non-associative Algebras Introduction

  • Construction A Non-associative Algebras

    Introduction to Space-Time Coding

    Frederique Oggierfrederique@ntu.edu.sg

    Division of Mathematical SciencesNanyang Technological University, Singapore

    Noncommutative Rings and their Applications V, Lens, 12-15June 2017

  • Construction A Non-associative Algebras

    Last Time

    1. A fully diverse space-time code is a family C of (square)complex matrices such that det(X X) 6= 0 when X 6= X.

    2. Division algebras whose elements can be represented asmatrices satisfy full diversity by definition.

    1. For coding for MIMO slow fading channels, joint design of aninner and outer code.

    2. The outer code is a coset code, which addresses the problemof codes over matrices.

    3. Connection between codes over matrices and codes over finitefields.

  • Construction A Non-associative Algebras

    Last Time

    1. A fully diverse space-time code is a family C of (square)complex matrices such that det(X X) 6= 0 when X 6= X.

    2. Division algebras whose elements can be represented asmatrices satisfy full diversity by definition.

    1. For coding for MIMO slow fading channels, joint design of aninner and outer code.

    2. The outer code is a coset code, which addresses the problemof codes over matrices.

    3. Connection between codes over matrices and codes over finitefields.

  • Construction A Non-associative Algebras

    Outline

    Construction AThe non-commutative case

    Non-associative Algebras

  • Construction A Non-associative Algebras

    Construction A

    Let : ZN 7 FN2 be the reductionmodulo 2 componentwise.

    Let C FN2 be an (N, k) linearbinary code.

    Then 1(C ) is a lattice.

    Let p be a primitive pth root ofunity, p a prime.

    Let : Z[p]N 7 FNp be thereduction componentwise modulothe prime ideal p = (1 p).

    Then 1(C ) is a lattice, when Cis an (N, k) linear code over Fp.

    In particular, p = 2 yields thebinary Construction A.

    What about a Construction A from division algebras?

  • Construction A Non-associative Algebras

    Construction A

    Let : ZN 7 FN2 be the reductionmodulo 2 componentwise.

    Let C FN2 be an (N, k) linearbinary code.

    Then 1(C ) is a lattice.

    Let p be a primitive pth root ofunity, p a prime.

    Let : Z[p]N 7 FNp be thereduction componentwise modulothe prime ideal p = (1 p).

    Then 1(C ) is a lattice, when Cis an (N, k) linear code over Fp.

    In particular, p = 2 yields thebinary Construction A.

    What about a Construction A from division algebras?

  • Construction A Non-associative Algebras

    Construction A

    Let : ZN 7 FN2 be the reductionmodulo 2 componentwise.

    Let C FN2 be an (N, k) linearbinary code.

    Then 1(C ) is a lattice.

    Let p be a primitive pth root ofunity, p a prime.

    Let : Z[p]N 7 FNp be thereduction componentwise modulothe prime ideal p = (1 p).

    Then 1(C ) is a lattice, when Cis an (N, k) linear code over Fp.

    In particular, p = 2 yields thebinary Construction A.

    What about a Construction A from division algebras?

  • Construction A Non-associative Algebras

    Ingredients

    A p

    K OK pOK

    F OF p

    Q Z p

    Let K/F be a cyclic number field extension ofdegree n, and rings of integers OK and OF .Consider the cyclic division algebra

    A = K Ke Ken1

    where en = u OF , and ek = (k)e for k K . Let be its natural order

    = OK OKe OKen1.

    Let p be a prime ideal of OF so that p is atwo-sided ideal of .

  • Construction A Non-associative Algebras

    Ingredients

    A p

    K OK pOK

    F OF p

    Q Z p

    Let K/F be a cyclic number field extension ofdegree n, and rings of integers OK and OF .Consider the cyclic division algebra

    A = K Ke Ken1

    where en = u OF , and ek = (k)e for k K .

    Let be its natural order

    = OK OKe OKen1.

    Let p be a prime ideal of OF so that p is atwo-sided ideal of .

  • Construction A Non-associative Algebras

    Ingredients

    A p

    K OK pOK

    F OF p

    Q Z p

    Let K/F be a cyclic number field extension ofdegree n, and rings of integers OK and OF .Consider the cyclic division algebra

    A = K Ke Ken1

    where en = u OF , and ek = (k)e for k K . Let be its natural order

    = OK OKe OKen1.

    Let p be a prime ideal of OF so that p is atwo-sided ideal of .

  • Construction A Non-associative Algebras

    Ingredients

    A p

    K OK pOK

    F OF p

    Q Z p

    Let K/F be a cyclic number field extension ofdegree n, and rings of integers OK and OF .Consider the cyclic division algebra

    A = K Ke Ken1

    where en = u OF , and ek = (k)e for k K . Let be its natural order

    = OK OKe OKen1.

    Let p be a prime ideal of OF so that p is atwo-sided ideal of .

  • Construction A Non-associative Algebras

    Skew-polynomial Rings

    Given a ring S with a group acting on it, theskew-polynomial ring S [x ;] is the set of polynomialss0 + s1x + . . .+ snx

    n, si S for i = 0, . . . , n, with xs = (s)xfor all s S .

    Lemma. There is an Fpf -algebra isomorphism between /pand the quotient of (OK/pOK )[x ;] by the two-sided idealgenerated by xn u.

  • Construction A Non-associative Algebras

    Skew-polynomial Rings

    Given a ring S with a group acting on it, theskew-polynomial ring S [x ;] is the set of polynomialss0 + s1x + . . .+ snx

    n, si S for i = 0, . . . , n, with xs = (s)xfor all s S .

    Lemma. There is an Fpf -algebra isomorphism between /pand the quotient of (OK/pOK )[x ;] by the two-sided idealgenerated by xn u.

  • Construction A Non-associative Algebras

    Quotients

    p /p

    OK p pOK

    OF OF p

    Z p Z/pZ

    There is an Fpf -algebra isomorphism

    : /p = (OK/pOK )[x ;]/(xn u).

    If p is inert, OK/pOK is a finite field

  • Construction A Non-associative Algebras

    Quotients

    p /p

    OK p pOK

    OF OF p

    Z p Z/pZ

    There is an Fpf -algebra isomorphism

    : /p = (OK/pOK )[x ;]/(xn u).

    If p is inert, OK/pOK is a finite field

  • Construction A Non-associative Algebras

    Codes over Finite Fields

    /p Fnq

    OK/p FNpf

    Z/pZ FNp

    Let I be a left ideal of , I OF p. ThenI/p is an ideal of /p and (I/p) a leftideal of Fq[x ;]/(xn u).

    Let f Fq[x ;] be a polynomial of degree n. If(f ) is a two-sided ideal of Fq[x ;], then a-code consists of codewordsa = (a0, a1, . . . , an1), where a(x) are leftmultiples of a right divisor g of f .

    Using : /p = Fq[x ;]/(xn u), for everyleft ideal I of , we get a -code C = (I/p)over Fq.

    [ D. Boucher and F. Ulmer, Coding with skew polynomial rings]

  • Construction A Non-associative Algebras

    Codes over Finite Fields

    /p Fnq

    OK/p FNpf

    Z/pZ FNp

    Let I be a left ideal of , I OF p. ThenI/p is an ideal of /p and (I/p) a leftideal of Fq[x ;]/(xn u).

    Let f Fq[x ;] be a polynomial of degree n. If(f ) is a two-sided ideal of Fq[x ;], then a-code consists of codewordsa = (a0, a1, . . . , an1), where a(x) are leftmultiples of a right divisor g of f .

    Using : /p = Fq[x ;]/(xn u), for everyleft ideal I of , we get a -code C = (I/p)over Fq.

    [ D. Boucher and F. Ulmer, Coding with skew polynomial rings]

  • Construction A Non-associative Algebras

    Codes over Finite Fields

    /p Fnq

    OK/p FNpf

    Z/pZ FNp

    Let I be a left ideal of , I OF p. ThenI/p is an ideal of /p and (I/p) a leftideal of Fq[x ;]/(xn u).

    Let f Fq[x ;] be a polynomial of degree n. If(f ) is a two-sided ideal of Fq[x ;], then a-code consists of codewordsa = (a0, a1, . . . , an1), where a(x) are leftmultiples of a right divisor g of f .

    Using : /p = Fq[x ;]/(xn u), for everyleft ideal I of , we get a -code C = (I/p)over Fq.

    [ D. Boucher and F. Ulmer, Coding with skew polynomial rings]

  • Construction A Non-associative Algebras

    Codes over Finite Fields

    /p Fnq

    OK/p FNpf

    Z/pZ FNp

    Let I be a left ideal of , I OF p. ThenI/p is an ideal of /p and (I/p) a leftideal of Fq[x ;]/(xn u).

    Let f Fq[x ;] be a polynomial of degree n. If(f ) is a two-sided ideal of Fq[x ;], then a-code consists of codewordsa = (a0, a1, . . . , an1), where a(x) are leftmultiples of a right divisor g of f .

    Using : /p = Fq[x ;]/(xn u), for everyleft ideal I of , we get a -code C = (I/p)over Fq.

    [ D. Boucher and F. Ulmer, Coding with skew polynomial rings]

  • Construction A Non-associative Algebras

    Codes over Finite Rings

    /p (OK/pOK )n

    OK/p (OK/pOK )N

    Z/pZ FNp

    Let g(x) be a right divisor of xn u. The ideal(g(x))/(xn u) is an OK/pOK -module,isomorphic to a submodule of (OK/pOK )n. Itforms a -constacyclic code of length n anddimension k = n degg(x), consisting ofcodewords a = (a0, a1, . . . , an1), where a(x) areleft multiples of g(x).

    A parity check polynomial is computed. A dual code is defined.

    [ Ducoat-O., On Skew Polynomial Codes and Lattices from Quotients of CyclicDivision Algebras]

  • Construction A Non-associative Algebras

    Codes over Finite Rings

    /p (OK/pOK )n

    OK/p (OK/pOK )N

    Z/pZ FNp

    Let g(x) be a right divisor of xn u. The ideal(g(x))/(xn u) is an OK/pOK -module,isomorphic to a submodule of (OK/pOK )n. Itforms a -constacyclic code of length n anddimension k = n degg(x), consisting ofcodewords a = (a0, a1, . . . , an1), where a(x) areleft multiples of g(x).

    A parity check polynomial is computed. A dual code is defined.