The Central Dogma of Genetics Or the Coding Theory Behind it Artur - PowerPoint PPT Presentation

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 1 / 24 PPS 2014, Oct 17th

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

Introduction to Coding Theory 1 Linear Codes and Related Codes 2 Group Ring Codes of extra-special Groups 3 Orthogonal Array’s and Codes 4 Artur Schäfer ( University of St. Andrews ) Coding Theory 3 / 24 PPS 2014, Oct 17th

Contents Introduction to Coding Theory 1 Linear Codes and Related Codes 2 Group Ring Codes of extra-special Groups 3 Orthogonal Array’s and Codes 4 Artur Schäfer ( University of St. Andrews ) Coding Theory 4 / 24 PPS 2014, Oct 17th

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 ( x 1 , x 2 ) between two n -tuples is the number of coordinates, where x 1 and x 2 do not coincide. The Hamming distances for C are { 3 , 3 , 3 } . Artur Schäfer ( University of St. Andrews ) Coding Theory 5 / 24 PPS 2014, Oct 17th

Definition The minimum distance of a code C is min { d ( x 1 , x 2 ) : x 1 , x 2 ∈ 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 6 / 24 PPS 2014, Oct 17th

Goal of Coding Theory Given n , d and q . Find a code C with M as big as possible! Reason: A code with d > 2 t + 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 7 / 24 PPS 2014, Oct 17th

Contents Introduction to Coding Theory 1 Linear Codes and Related Codes 2 Group Ring Codes of extra-special Groups 3 Orthogonal Array’s and Codes 4 Artur Schäfer ( University of St. Andrews ) Coding Theory 8 / 24 PPS 2014, Oct 17th

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 ( c 0 , ..., c n − 2 , c n − 1 ) ∈ C ⇒ ( c n − 1 , c 0 , ..., c n − 2 ) ∈ C . n − 1 � 3 If C is cyclic then, via ( c 0 , ..., c n − 1 ) �→ c i x i , we can identify C i = 0 as an ideal C � F [ x ] / ( x n − 1 ) . Artur Schäfer ( University of St. Andrews ) Coding Theory 9 / 24 PPS 2014, Oct 17th

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 � 1 � α 2 0 , with F × Let C ≤ F 3 4 be given by G = 4 = � α � . 0 1 α ( Reed-Solomon ). C is cyclic and satisfies ( c 1 , ..., c n − 1 , c n ) ∈ C ⇒ ( c n , c 1 , ..., c n − 1 ) ∈ C . Other cyclic codes: BCH-codes, binary Hamming-codes. Artur Schäfer ( University of St. Andrews ) Coding Theory 10 / 24 PPS 2014, Oct 17th

Alternative description Also, it is common to provide a parity check matrix H . The code C is the kernel of this matrix. ⇒ GH T = 0. Beispiel � 1 � α 2 0 , with F × G = 4 = � α � . We get 0 1 α � � α 2 H = α 1 Artur Schäfer ( University of St. Andrews ) Coding Theory 11 / 24 PPS 2014, Oct 17th

Definition C ≤ F n a code n × n permutationmatrices ∼ Perm n ( F ) = = S n � F × � n ⋊ S n n × n monomial matrices ∼ Mon n ( F ) = = Perm ( C ) = { M ∈ Perm n ( F ) | C . M = C } Mon ( C ) = { M ∈ Mon n ( F ) | C . M = C } . If two codes C , D ≤ F n satisfy C . M = D , for M ∈ Mon n ( 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 12 / 24 PPS 2014, Oct 17th

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 C n = � g � , we have the following correspondence: I � F [ x ] / ( x n − 1 ) � C ≤ F n ↔ ↔ I � FC n � n � n c i x i c i g i ( c 1 , ..., c n ) ↔ ↔ i = 1 i = 1 ⇒ Consider codes of group rings FG Artur Schäfer ( University of St. Andrews ) Coding Theory 13 / 24 PPS 2014, Oct 17th

Definition Let G be a finite group, F a finite field and { g 1 , ..., g n } a basis of the vector space FG . Then, any ideal I of FG defines a code C ≤ F n by: ( a 1 , a 2 , ..., a n ) ∈ C ⇔ a 1 g 1 + a 2 g 2 + · · · + a n g n ∈ 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 14 / 24 PPS 2014, Oct 17th

Contents Introduction to Coding Theory 1 Linear Codes and Related Codes 2 Group Ring Codes of extra-special Groups 3 Orthogonal Array’s and Codes 4 Artur Schäfer ( University of St. Andrews ) Coding Theory 15 / 24 PPS 2014, Oct 17th

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 , ..., g p − 1 , t 2 , t 2 g , ..., t 2 g p − 1 , ..., t p 2 n , t p 2 n g , ..., t p 2 n g p − 1 � } � �� � � �� p elements Artur Schäfer ( University of St. Andrews ) Coding Theory 16 / 24 PPS 2014, Oct 17th

Theorem An extra-special group G has the structure of a symplectic vector space V with V = � a 1 , b 1 � ⊥ · · · ⊥ � a n , b n � . Corollary p 2 n p − 1 � � FG = e i FG ⊕ f k FG . ���� � �� � i = 1 k = 1 dim ( ··· )= p 2 n dim ( ··· )= 1 � �� � � �� � I I ⊥ Artur Schäfer ( University of St. Andrews ) Coding Theory 17 / 24 PPS 2014, Oct 17th

  p -times � �� �   1 · · · 1 0 · · · 0 0 · · · 0 · · · 0 · · · 0     0 · · · 0 1 · · · 1 0 · · · 0 · · · 0 · · · 0   ∈ F p 2 n × p 2 n + 1 . I =   0 · · · 0 0 · · · 0 1 · · · 1 · · · 0 · · · 0     . . . . ... . . . .   . . . . 0 · · · 0 0 · · · 0 0 · · · 0 · · · 1 · · · 1 ⇒ I is a semi-cyclic code, see repetition code .   1 0 0 · · · 0 − 1   0 1 0 · · · 0 − 1     ⇒ I ⊥ = ⊥ p 2 n 0 0 1 · · · 0 − 1 .   i = 1   . . . . ... . . . .   . . . . 0 0 0 · · · 1 − 1 ⇒ I ⊥ is a semi-cyclic code. Artur Schäfer ( University of St. Andrews ) Coding Theory 18 / 24 PPS 2014, Oct 17th

Contents Introduction to Coding Theory 1 Linear Codes and Related Codes 2 Group Ring Codes of extra-special Groups 3 Orthogonal Array’s and Codes 4 Artur Schäfer ( University of St. Andrews ) Coding Theory 19 / 24 PPS 2014, Oct 17th

Definition A t − ( v , k , λ ) orthogonal array (OA) is a λ v t × k array whose entries are chosen from a set X with v points such that in every subset of t columns of the array, every t -tuple of points of X appears in exactly λ rows. Beispiel   1 1 1   1 2 2     1 3 3     2 1 2     OA = 2 2 3     2 3 1     3 1 3     3 2 1 3 3 2 Artur Schäfer ( University of St. Andrews ) Coding Theory 20 / 24 PPS 2014, Oct 17th

Definition A Latin hypercube of order n and dimension m is Z m 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 21 / 24 PPS 2014, Oct 17th

  1 1 1 1   1 2 2 2     1 3 3 3     2 1 2 3     OA = 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 n m − 1 times. Artur Schäfer ( University of St. Andrews ) Coding Theory 22 / 24 PPS 2014, Oct 17th

New Stuff! Definition A Hamming graph H S ( m , n ) is a graph with vertices Z m 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 n m − k of H S ( m , n ) are orthogonal arrays. ⇒ finding mutually orthogonal Latin squares is equivalent to finding maximal cliques! Artur Schäfer ( University of St. Andrews ) Coding Theory 23 / 24 PPS 2014, Oct 17th

Thank You for Your Attention Artur Schäfer ( University of St. Andrews ) Coding Theory 24 / 24 PPS 2014, Oct 17th