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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

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

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.