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The Central Dogma of Genetics Or the Coding Theory Behind it Artur Schfer University of St. Andrews PPS 2014, Oct 17th Quiz: What is the next number? 3, 9, 21, 45, 93,... Artur Schfer ( University of St. Andrews ) Coding Theory 1 / 24

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The Central Dogma of Genetics

Or the Coding Theory Behind it Artur Schäfer

University of St. Andrews

PPS 2014, Oct 17th Quiz: What is the next number? 3, 9, 21, 45, 93,...

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 1 / 24

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Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 2 / 24

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1

Introduction to Coding Theory

2

Linear Codes and Related Codes

3

Group Ring Codes of extra-special Groups

4

Orthogonal Array’s and Codes

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 3 / 24

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Contents

1

Introduction to Coding Theory

2

Linear Codes and Related Codes

3

Group Ring Codes of extra-special Groups

4

Orthogonal Array’s and Codes

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 4 / 24

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Definition

Let A be an alphabet. A code C of length n over the alphabet A is a set of n-tuples with entries in A.

Beispiel

Using A = {0, 1, 2}, then C = {000, 121, 212} is a ternary code of length 3. Recall: The Hamming distance d(x1, x2) between two n-tuples is the number of coordinates, where x1 and x2 do not coincide. The Hamming distances for C are {3, 3, 3}.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 5 / 24

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Definition

The minimum distance of a code C is min{d(x1, x2) : x1, x2 ∈ C}. To each code we can attach parameters (n, M, d, q). n = length of C M = # elements in C d = minimum distance of C q = size of A

Beispiel

C = {000, 121, 212} is a (3, 3, 3, 3)−code.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 6 / 24

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Goal of Coding Theory

Given n, d and q.

Find a code C with M as big as possible!

Reason: A code with d > 2t + 1, can correct t errors, for any t. A code with big M is more useful than for small M.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 7 / 24

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Contents

1

Introduction to Coding Theory

2

Linear Codes and Related Codes

3

Group Ring Codes of extra-special Groups

4

Orthogonal Array’s and Codes

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 8 / 24

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Definition

1 Let F be a finite field and n a non negative integer. A linear code C

is a subspace C ≤ F n.

2 Let C be a linear code. C is cyclic if

(c0, ..., cn−2, cn−1) ∈ C ⇒ (cn−1, c0, ..., cn−2) ∈ C.

3 If C is cyclic then, via (c0, ..., cn−1) →

n−1

  • i=0

cixi, we can identify C as an ideal C F[x]/(xn − 1).

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 9 / 24

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Since the code is subspace we use a matrix G to describe it, where the rows of G form a basis of this space.

Beispiel

Let C ≤ F3

4 be given by G =

1 α2 1 α

  • , with F×

4 = α.

(Reed-Solomon). C is cyclic and satisfies (c1, ..., cn−1, cn) ∈ C ⇒ (cn, c1, ..., cn−1) ∈ C. Other cyclic codes: BCH-codes, binary Hamming-codes.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 10 / 24

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Alternative description

Also, it is common to provide a parity check matrix H. The code C is the kernel of this matrix. ⇒ GHT = 0.

Beispiel

G = 1 α2 1 α

  • , with F×

4 = α. We get

H =

  • α2

α 1

  • Artur Schäfer (University of St. Andrews)

Coding Theory PPS 2014, Oct 17th 11 / 24

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Definition

C ≤ F n a code Permn(F) = n × n permutationmatrices ∼ = Sn Monn(F) = n × n monomial matrices ∼ =

  • F ×n ⋊ Sn

Perm(C) = {M ∈ Permn(F) | C.M = C} Mon(C) = {M ∈ Monn(F) | C.M = C}. If two codes C, D ≤ F n satisfy C.M = D, for M ∈ Monn(F), then they are equivalent.

Lemma

Equivalent codes have the same parameters. C = {000, 121, 212} and D = {000, 111, 222} are equivalent (3, 3, 3, 3)−codes.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 12 / 24

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Cyclic Codes are the most Important ones

So, far cyclic codes have the best parameters (n, M, d, q) for practical use so far. → Reed-Solomon codes for CD’s.

Does a generalization of cyclic codes lead to better codes?

For a cyclic code C a monogenetic inverse semigroup Cn = g, we have the following correspondence: C ≤ F n ↔ I F[x]/(xn − 1) ↔

  • I FCn

(c1, ..., cn) ↔

n

  • i=1

cixi ↔

n

  • i=1

cigi ⇒ Consider codes of group rings FG

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 13 / 24

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Definition

Let G be a finite group, F a finite field and {g1, ..., gn} a basis of the vector space FG. Then, any ideal I of FG defines a code C ≤ F n by: (a1, a2, ..., an) ∈ C ⇔ a1g1 + a2g2 + · · · + angn ∈ I. Any code equivalent to C, for any ideal is a G-code.

Definition

If G is abelian/cyclic, then we say the G-code is abelian/cyclic.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 14 / 24

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Contents

1

Introduction to Coding Theory

2

Linear Codes and Related Codes

3

Group Ring Codes of extra-special Groups

4

Orthogonal Array’s and Codes

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 15 / 24

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We consider semi-simple group rings over a field with sufficiently many roots of unity.

Definition

An extra-special group is a group G, with Z(G) = G ′ = {1} (cetre subgroup, commutator subgroup).

Cosets

We get the coset decomposition G/G ′ G = {1, g, ..., gp−1, t2, t2g, ..., t2gp−1

  • p elements

, ..., tp2n, tp2ng, ..., tp2ngp−1

  • }

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 16 / 24

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Theorem

An extra-special group G has the structure of a symplectic vector space V with V = a1, b1 ⊥ · · · ⊥ an, bn.

Corollary

FG =

p2n

  • i=1

eiFG

  • dim(··· )=1
  • I

⊕

p−1

  • k=1

fkFG

dim(··· )=p2n

  • I ⊥

.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 17 / 24

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I =         

p-times

1 · · · 1 0 · · · 0 0 · · · 0 · · · 0 · · · 0 0 · · · 0 1 · · · 1 0 · · · 0 · · · 0 · · · 0 0 · · · 0 0 · · · 0 1 · · · 1 · · · 0 · · · 0 . . . . . . . . . ... . . . 0 · · · 0 0 · · · 0 0 · · · 0 · · · 1 · · · 1          ∈ F p2n×p2n+1. ⇒ I is a semi-cyclic code, see repetition code. ⇒ I ⊥ =⊥p2n

i=1

       1 · · · −1 1 · · · −1 1 · · · −1 . . . . . . . . . ... . . . · · · 1 −1        . ⇒ I ⊥ is a semi-cyclic code.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 18 / 24

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Contents

1

Introduction to Coding Theory

2

Linear Codes and Related Codes

3

Group Ring Codes of extra-special Groups

4

Orthogonal Array’s and Codes

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 19 / 24

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Definition

A t − (v, k, λ) orthogonal array (OA) is a λvt × k array whose entries are chosen from a set X with v points such that in every subset of t columns

  • f the array, every t-tuple of points of X appears in exactly λ rows.

Beispiel

OA =               1 1 1 1 2 2 1 3 3 2 1 2 2 2 3 2 3 1 3 1 3 3 2 1 3 3 2              

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 20 / 24

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Definition

A Latin hypercube of order n and dimension m is Zm

n , where each tuple is

labelled with an additional entry from 1, .., n such that each line contains an entry only once. Latin hypercubes are orthogonal arrays, but not every OA is a Latin hypercube.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 21 / 24

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OA =               1 1 1 1 1 2 2 2 1 3 3 3 2 1 2 3 2 2 3 1 2 3 1 2 3 1 3 2 3 2 1 3 3 3 2 1              

Definition

Two LHC are mutually orthogonal, if their labels form tuples where each tuple appears exactly nm−1 times.

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 22 / 24

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New Stuff!

Definition

A Hamming graph HS(m, n) is a graph with vertices Zm

n , where two

vertices have an edge if their Hamming distance is in the set S ⊆ {1, ..., m}.

Proposition

If S = {k + 1, ..., m}, then maximal cliques of size nm−k of HS(m, n) are

  • rthogonal arrays.

⇒ finding mutually orthogonal Latin squares is equivalent to finding maximal cliques!

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 23 / 24

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Thank You for Your Attention

Artur Schäfer (University of St. Andrews) Coding Theory PPS 2014, Oct 17th 24 / 24