Monoids and Maximal Codes Fabio Burderi Dipartimento di Matematica e Informatica Universit` a degli studi di Palermo, burderi@math.unipa.it WORDS, Sept. 12-16 2011, Prague logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 1
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . Attention !! A code X over A is a subset of A + . The words of X are called code words , the elements of X + messages . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . Attention !! A code X over A is a subset of A + . The words of X are called code words , the elements of X + messages . If w ∈ A ∗ , a factorization of w is a sequence of words ( v i ) 1 ≤ i ≤ s such that w = v 1 v 2 · · · v s . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . Attention !! A code X over A is a subset of A + . The words of X are called code words , the elements of X + messages . If w ∈ A ∗ , a factorization of w is a sequence of words ( v i ) 1 ≤ i ≤ s such that w = v 1 v 2 · · · v s . If X is a code, a relation between code words is a pair of factorizations x 1 x 2 · · · x s = y 1 y 2 · · · y t into code words of a same message w ∈ X + ; the relation is said non-trivial if the factorizations are distinct. logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . Attention !! A code X over A is a subset of A + . The words of X are called code words , the elements of X + messages . If w ∈ A ∗ , a factorization of w is a sequence of words ( v i ) 1 ≤ i ≤ s such that w = v 1 v 2 · · · v s . If X is a code, a relation between code words is a pair of factorizations x 1 x 2 · · · x s = y 1 y 2 · · · y t into code words of a same message w ∈ X + ; the relation is said non-trivial if the factorizations are distinct. We say that the relation x 1 x 2 · · · x s = y 1 y 2 · · · y t is prime if for all i < s and for all j < t one has x 1 x 2 · · · x i � = y 1 y 2 · · · y j . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
Let A be an alphabet. Let A ∗ denote the free monoid generated by A , and let A + = A ∗ \{ ε } . Attention !! A code X over A is a subset of A + . The words of X are called code words , the elements of X + messages . If w ∈ A ∗ , a factorization of w is a sequence of words ( v i ) 1 ≤ i ≤ s such that w = v 1 v 2 · · · v s . If X is a code, a relation between code words is a pair of factorizations x 1 x 2 · · · x s = y 1 y 2 · · · y t into code words of a same message w ∈ X + ; the relation is said non-trivial if the factorizations are distinct. We say that the relation x 1 x 2 · · · x s = y 1 y 2 · · · y t is prime if for all i < s and for all j < t one has x 1 x 2 · · · x i � = y 1 y 2 · · · y j . A relation w = x 1 x 2 · · · x s = y 1 y 2 · · · y t , can be univocally factorized into prime relations. logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 2
A code X is said to be uniquely decipherable ( UD ) if there are not non-trivial relations on X . Every message has an unique factorization into code words: x 1 x 2 · · · x n = y 1 y 2 · · · y m , x i , y j ∈ X implies n = m and x 1 = y 1 , . . . , x n = y n . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 3
A code X is said to be uniquely decipherable ( UD ) if there are not non-trivial relations on X . Every message has an unique factorization into code words: x 1 x 2 · · · x n = y 1 y 2 · · · y m , x i , y j ∈ X implies n = m and x 1 = y 1 , . . . , x n = y n . X = A 2 = { 00 , 01 , 10 , 11 } , A = { 0 , 1 } , Example 1 z = 0100101011 logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 3
A code X is said to be uniquely decipherable ( UD ) if there are not non-trivial relations on X . Every message has an unique factorization into code words: x 1 x 2 · · · x n = y 1 y 2 · · · y m , x i , y j ∈ X implies n = m and x 1 = y 1 , . . . , x n = y n . X = A 2 = { 00 , 01 , 10 , 11 } , A = { 0 , 1 } , Example 1 z = 0100101011 = 01 · 00 · 10 · 10 · 11 logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 3
A code X is said to be uniquely decipherable ( UD ) if there are not non-trivial relations on X . Every message has an unique factorization into code words: x 1 x 2 · · · x n = y 1 y 2 · · · y m , x i , y j ∈ X implies n = m and x 1 = y 1 , . . . , x n = y n . X = A 2 = { 00 , 01 , 10 , 11 } , A = { 0 , 1 } , Example 1 z = 0100101011 = 01 · 00 · 10 · 10 · 11 X = { 0 , 01 , 10 } Example 2 z = 010 logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 3
A code X is said to be uniquely decipherable ( UD ) if there are not non-trivial relations on X . Every message has an unique factorization into code words: x 1 x 2 · · · x n = y 1 y 2 · · · y m , x i , y j ∈ X implies n = m and x 1 = y 1 , . . . , x n = y n . X = A 2 = { 00 , 01 , 10 , 11 } , A = { 0 , 1 } , Example 1 z = 0100101011 = 01 · 00 · 10 · 10 · 11 X = { 0 , 01 , 10 } Example 2 z = 010 = 0 · 10 = 01 · 0 logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 3
Let X be a code and let P = { X i | i ∈ I } be a partition of X i.e. : � i ∈ I X i = X and X i ∩ X j = ∅ , iff i � = j . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 4
Let X be a code and let P = { X i | i ∈ I } be a partition of X i.e. : � i ∈ I X i = X and X i ∩ X j = ∅ , iff i � = j . A P - factorization of a message w ∈ X + is a factorization w = z 1 z 2 · · · z t where: for each i z i ∈ X + for some X k ∈ P k , if t > 1, z i ∈ X + ∈ X + k ⇒ z i +1 / (1 ≤ i ≤ t − 1). k , logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 4
Example 3 X = { 00 , 11 , 000 , 111 } = { x 1 , x 2 , x 3 , x 4 } , P = { X 1 , X 2 , } , X 1 = { 00 , 11 } , X 2 = { 000 , 111 } . Let w = 1100000111 ∈ X + , w = 11 · 00 · 000 · 111 logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 5
Example 3 X = { 00 , 11 , 000 , 111 } = { x 1 , x 2 , x 3 , x 4 } , P = { X 1 , X 2 , } , X 1 = { 00 , 11 } , X 2 = { 000 , 111 } . Let w = 1100000111 ∈ X + , w = 11 · 00 · 000 · 111 w = z 1 z 2 = (11 · 00)(000 · 111) logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 5
Example 3 X = { 00 , 11 , 000 , 111 } = { x 1 , x 2 , x 3 , x 4 } , P = { X 1 , X 2 , } , X 1 = { 00 , 11 } , X 2 = { 000 , 111 } . Let w = 1100000111 ∈ X + , w = 11 · 00 · 000 · 111 w = z 1 z 2 = (11 · 00)(000 · 111) w = 11 · 000 · 00 · 111 w = u 1 u 2 u 3 u 4 = (11)(000)(00)(111) z 1 z 2 and u 1 u 2 u 3 u 4 are P − factorizzations of z . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 5
Example 3 X = { 00 , 11 , 000 , 111 } = { x 1 , x 2 , x 3 , x 4 } , P = { X 1 , X 2 , } , X 1 = { 00 , 11 } , X 2 = { 000 , 111 } . Let w = 1100000111 ∈ X + , w = 11 · 00 · 000 · 111 w = z 1 z 2 = (11 · 00)(000 · 111) w = 11 · 000 · 00 · 111 w = u 1 u 2 u 3 u 4 = (11)(000)(00)(111) z 1 z 2 and u 1 u 2 u 3 u 4 are P − factorizzations of z . The partition P is called a coding partition if any element w ∈ X + has a unique P - factorization , i.e. if w = z 1 z 2 · · · z s = u 1 u 2 · · · u t , z 1 z 2 · · · z s , u 1 u 2 · · · u t with P - factorizations of w , then: s = t and z i = u i for i = 1 , . . . , s . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 5
Example 3 X = { 00 , 11 , 000 , 111 } , P = { X 1 , X 2 , } , X 1 = { 00 , 000 } , X 2 = { 11 , 111 } . w = 1100000111 = 11 · 00000 · 111 P is a coding partition of X . logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 6
Example 3 X = { 00 , 11 , 000 , 111 } , P = { X 1 , X 2 , } , X 1 = { 00 , 000 } , X 2 = { 11 , 111 } . w = 1100000111 = 11 · 00000 · 111 P is a coding partition of X . Let P = { X i | i ∈ I } be a partition of a code X . The partition P is a coding partition iff for every prime relation x 1 x 2 · · · x s = y 1 y 2 · · · y t , the code words x i , y j belong to the same component of the partition. logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 6
Example 3 X = { 00 , 11 , 000 , 111 } , P = { X 1 , X 2 , } , X 1 = { 00 , 000 } , X 2 = { 11 , 111 } . w = 1100000111 = 11 · 00000 · 111 P is a coding partition of X . Let P = { X i | i ∈ I } be a partition of a code X . The partition P is a coding partition iff for every prime relation x 1 x 2 · · · x s = y 1 y 2 · · · y t , the code words x i , y j belong to the same component of the partition. A code X is called ambiguous if it is not UD . A code is called totally ambiguous ( TA ) if | X | > 1 and the only coding partition is the trivial partition: P = { X } . X = { 0 , 01 , 10 } . Example 2 The word w = 010 ∈ X + has two factorizations : w = 0 · 10 = 01 · 0. X is a totally ambiguous code. logo WORDS ’11 Prague Fabio Burderi Monoids and Maximal Codes 6
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