A two dimensional non-overlapping code over a q-ary alphabet

Transcription

1 A two dimensional non-overlapping code over a q-ary alphabet Antonio Bernini Dipartimento di Matematica e Informatica, Università di Firenze Convegno GNCS Montecatini Terme, Febbraio 2018 Progetto di Ricerca 2017: Codici di stringhe e matrici non sovrapponibili Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

2 Background on strings Bifix-free strings: description The string has the bifix The string has not any bifix: it is said bifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

3 Background on strings Bifix-free strings: description Bifix-free strings are collected in a set: Definition Denote BF q n the set of n-length bifix-free strings on a q-ary alphabet. Theorem (Nielsen, 1973) BF q 1 = q BF q 2n = q BF q 2n 1 BF n q BF q 2n+1 = q BF q 2n Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

4 Background on strings Cross-bifix-free strings: description The two strings in BF , admit a cross-bifix: The two strings in BF , are cross-bifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

5 Background on strings Definition A set of strings is said cross-bifix-free if each string is bifix-free and if any two strings are cross-bifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

6 Background on strings A cross-bifix-free set Definition Fixed 2 k n 2, denote S (k) n,q the set of strings s[1]s[2]... s[n] of length n on the alphabet Σ = {0, 1,..., q 1} of the form s[1] s[2] s[k] s[k + 1] }{{}}{{} = 1 k 1 s[k + 2] s[n 1] }{{} avoiding 1 k s[n] }{{} 1 Example: if n = 7, k = 2 and q = 2 we obtain: S (2) 7,2 = { , , , , } while S (3) 7,2 = { , , , } Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

7 Background on strings A cross-bifix-free set Theorem (Chee, Kiah, Purkayastha, Wang, 2013) S n,q (k) is a non-expandable cross-bifix-free set on BFn q. Moreover, S n,q (k) = (q 1) 2 f (k) n k 2,q, where f (k) n,q { u Σ n u avoids 1 k} = q n (q 1) k j=1 f (k) n j,q se 0 n < k se n k Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

8 Extension to the bidimensional case: a first approach Bibifix-free square matrices: description The matrix ( ) 1 1 has the bibifix. It is a submatrix which appears at same time in two 1 0 opposite corners on the main diagonal. The matrix has not any bibifix: it is said bibifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

9 Extension to the bidimensional case: a first approach Bibifix-free square matrices: description Bibifix-free square matrices are collected in a set: Definition Denote BBF q n the set of bibifix-free matrices of dimension n n over a finite alphabet Σ = {0, 1,..., q 1} Theorem (Barcucci, Bernini, Bilotta, Pinzani, 2015) BBF q 0 = 1, BBFn q = q 2n 1 BBF q n 1, n 1, n odd, BBFn q = q 2n 1 BBF q n 1 q n2 2 BBF q n, n 1, n even. 2 Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

10 Cross-bibifix-free square matrices Description The matrices of BBF , overlap (cross each other) along the direction of the main diagonal: Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

11 Cross-bibifix-free square matrices Description The matrices of BBF , do not overlap along the direction of the main diagonal: they are said cross-bibifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

12 Cross-bibifix-free square matrices Definition A set of square matrices is said cross-bibifix-free if each matrix is bibifix-free and if any two matrices are cross-bibifix-free. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

13 Cross-bibifix-free square matrices A set of cross-bibifix-free matrices Definition u[1] CBBFn q. = u[2] u[n] u Sn q, Σ The set Sn q is such that Sn q (k) = max S n,q k=2,...,n 2 Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

14 Cross-bibifix-free square matrices A set of cross-bibifix-free matrices Definition u[1] CBBFn q. = u[2] u[n] u Sn q, Σ The set Sn q is such that Sn q (k) = max S n,q k=2,...,n 2 Proposition (Barcucci, Bernini, Bilotta, Pinzani, 2015) CBBF q n is a non-expandable cross-bibifix-free set on BBF q n. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

15 Non-overlapping matrices The matrices , can overlap in several ways: Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

16 Non-overlapping matrices The matrices , do not overlap along any direction of the plane: they are said non-overlapping. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

17 A set of non-overlapping binary matrices: definition Fixed 3 k n (k) 2, define S m n the set of matrices m n of the form M = (M i,j ) = 1 2 k 1 k k + 1 n k n k + 1 n k + 2 n 1 n such that the strings: 0M 1,k+1... M 1,n k 1 (red substring in the first row); M j,1... M j,n 1 0 for j = 2,..., m 1 (red strings in the row from the second to the second to last one); M m,k+1... M m,n k (red substring in the last row), avoid 0 k and 1 k Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

18 A set of non-overlapping binary matrices Definition A set of matrices is called non-overlapping if each matrix is self non-overlapping and for any two matrices in the set they are non-overlapping matrices. It is easily seen that each matrix of S (k) m n is self non-overlapping. Theorem (Barcucci, Bernini, Bilotta, Pinzani) The set S (k) m n is non-overlapping, for each k with 3 k n 2, m 2 and n 2k. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

19 A set of non-overlapping binary matrices Sketch of proof: If a matrix B could be overlapped on a matrix A, then a central row of A should contain a forbidden pattern: A = = B If the movement of B on A involved a fixed entry on the frame, some problem would appear in that entry: A = = B Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

20 A set of non-overlapping binary matrices: enumeration 1 2 k 1 k k + 1 n k n k + 1 n k + 2 n 1 n Denote B (k) n = { u {0, 1} n u avoids 0 k and 1 k} : first row: enumerated by r (k) (k) n 2k+2 where r n { } R n (k) = u B n (k) u 1 = 0, u n = 1 ; central { rows: enumerated } by z n (k) Z n (k) = u B n (k) u n = 0 ; where z (k) n = R n (k) and = Z n (k) and last row: enumerated by b (k) n 2k where b(k) n = B n (k). ( S m n (k) = r (k) n 2k+2 z (k) n ) m 2 b (k) n 2k Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

21 Strings in B (k), Z (k) and R (k) n n Recurrences for b n (k), z n (k) and r n (k) b (k) n = z n (k) = 2 n if 0 n k 1 k 1 j=1 b (k) n j if n k 1 if n = 0 b (k) n / 2 if n 1 1 if n = 0 n r (k) n = 0 if n = 1 2 n 2 if 2 n k k 1 ( ) z (k) n j r (k) n j j=1 if n k + 1 Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

22 Relation with k-generalized Fibonacci numbers f (k) n Revised recurrences f (k) n = 2 n if 0 n k 1 k j=1 b n (k) = z n (k) = r n (k) = f (k) n j if n k 1 if n = 0 2f (k 1) n 1 if n 1 1 if n = 0 f (k 1) n 1 if n 1 1 if n = 0 f (k 1) n 1 +d (k) n 2 if n 1 k-generalized Fibonacci numbers with d n (k) = 1 if (n mod k) = 0 1 if (n mod k) = 1 0 if (n mod k) 2 Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

23 Relation with k-generalized Fibonacci numbers Generating functions b (k) (x) = n 0 b n (k) x n = 1 + 2xf (k 1) (x) z (k) (x) = n 0 z n (k) x n = 1 + xf (k 1) (x) r (k) (x) = n 0 r (k) n x n = xf (k 1) (x) + d (k) (x) 1 2 where d (k) (x) = n 0 d (k) n x n = 1 k i=1 x i and f (k) (x) = f n (k) x n = n 0 k i=1 k i=1 x i x i Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

24 Relation with k-generalized Fibonacci numbers Cardinality The number S (k) (k) m n can now be expressed in term of f n : S (k) m n = ( f (k 1) 2k 1 ) m 2 if n = 2k ( ) ( f (k 1) n 2k 1 f (k 1) n 2k+1 + d (k) n 2k+2 f (k 1) n 1 ) m 2 if n > 2k ( ( In particular, if k = 3, using f n (2) = possible to obtain a closed formula for S (3) m n 2 ) n+2 ( 1 ) ) n for n 0, it is Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

25 A possible extension to a q-ary alphabet Let Σ = {0, 1, 2,..., q 1}, P Σ, {P 1, P 2 } a partition of P. W.l.o.g assume P = {0, 1, 2,..., p 1}, with 2 p q and P 1 = {0, 1,..., j 1}, P 2 = {j, j + 1,..., p 1}. Definition For each (α, β) P 1 P 2, we construct the set S (k,p) m n,q (α, β) of matrices A = (a i,j ) satisfying the following conditions: A 1 = α k 1 βw 1 αβ k 1, where v 1 = βw 1 α is a q-ary string of length n 2k + 2 avoiding all the patterns 0 k, 1 k, 2 k,..., (p 1) k ; for i = 2,..., m 1, A i = w i β = v i, where v i is a q-ary string of length n avoiding all the patterns 0 k, 1 k, 2 k,..., (p 1) k ; A m = α k v m β k, where v m is a q-ary string of length n 2k avoiding all the patterns 0 k, 1 k, 2 k,..., (p 1) k. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

26 Example: S (3,4) 4 9,6 (α, β) q = 6 (Σ = {0, 1, 2, 3, 4, 5}) alphabet p = 4 (P = {0, 1, 2, 3}) number of forbidden patterns P 1 = {0, 1}, P 2 = {2, 3} partition k = 3 length of forbidden patterns P 1 P 2 = {(0, 2), (0, 3), (1, 2), (1, 3)} (α, β) red (sub)strings avoid the pattern 000, 111, 222 and 333. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

27 Let S (k,p) m n,q = (α,β) S (k,p) m n,q (α, β) Proposition (Barcucci, Bernini, Bilotta, Pinzani, 2017) The set S (k,p) m n,q M m n,q is non-overlapping, for each k, p, m, n with 2 k n 2, 2 p q and m 2. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

28 Enumeration S (k,p) m n,q = (α,β) S (k,p) m n,q(α, β) Referring to the above definition: S (k,p) = r (k,p) where m n,q(α, β) r (k,p) n 2k+2,q counts the strings v 1; n 2k+2,q = j(p j) ( z (k,p) n,q z (k,p) n,q counts the strings v i, i = 2, 3,..., m; b (k,p) S m n,q(α, (k,p) β) ) m 2 b (k,p) n 2k,q n 2k,q counts the strings v m S (3,3) 4 9,q (1, 3) = Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

29 Recursive relations: not so useful b (k,p) n,q = q n if 0 n k 1 k 1 (q 1) i=1 b (k,p) n i,q + (q p)b(k,p) n k,q if n k. 0 if n = 0 z (k,p) n,q = q n 1 if 1 n k 1 k 1 ( ) b (k,p) n i,q z (k,p) n i,q i=1 if n k. 0 if n = 0, 1 r (k,p) n,q = q n 2 k 1 ( ) z (k,p) n i,q r (k,p) n i,q i=1 if 2 n k if n k + 1. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

30 Asymptotic behaviour (as n tends to + ) (Rational) Generating functions: b q (k,p) (x) = h(x)/g(x) z q (k,p) (x) = (h(x) 1)/g(x) r q (k,p) (x) = (h(x) 1) 2 /h(x)g(x) where g(x) = 1 (q 1)(x + x x j 1 ) (q p)x k and h(x) = (1 + x + x x k 1 ). It is possible to prove that the three denominators have the same unique zero x 0 (k, p, q) with smallest modulo (it is x 0 (k, p, q) < 1 for q 3) and multiplicity d 0 = 1. Therefore,... Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

31 Asymptotic behaviour... there exist three constants L 0, M 0, and N 0 such that (Expansion of rational functions) b (k,p) n,q L 0 z(k,p) n,q M 0 r n,q (k,p) N 0 Then, considering j = p 2, we have S (k,p) m n,q p p = 2 2 and, as n tends to +, ( ( r (k,p) n 2k+2,q ( 1 x 0 (k, p, q) z (k,p) n,q ) n ) m 2 b (k,p) Proposition S (k,p) p p ( m n,q L 0 M m N x 0 (k, p, q) n 2k,q ) ) nm 4k+2 Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

32 On the maximal size: S (k,p) m n,q = p p 2 2 ( ( r (k,p) n 2k+2,q z (k,p) n,q ) m 2 b (k,p) n 2k,q We are interested in the values of k and p giving the maximal value of the set when n is sufficiently large, once q is fixed. It is q qp(q 1) q(q k 1) p(q 1)(k 1) < 1 x 0 (k, p, q) < q p(q 1)k 1 (q 2) q k 1 (q 1) k q k 1 Recall that, for n sufficiently large, z (k,p) n,q M 0 ( 1 x 0 ) n ) 1. If k increases, the value 1/x 0 is expected to increase. So, z n has maximal size for k = n/2. 2. If p decreases, the value 1/x 0 is expected to increase. So, z n has maximal size for p = 2. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

33 On the maximal size: S (k,p) m n,q = p p 2 2 ( ( r (k,p) n 2k+2,q z (k,p) n,q ) m 2 b (k,p) n 2k,q ) Recall that ( ) n 2k+2 ( ) n 2k r (k,p) 1 n 2k+2,q N 0 and b (k,p) 1 n 2k,q x L 0 0 x 0 3. Different behaviour (unimodal behaviour) of b n 2k and r n 2k+2 with respect to z n depending on k (which appears in the exponent). 4. Same behaviour with respect to z n depending on p: maximal size for p = The coefficient p 2 p 2 is maximum if p = q. Summarizing: it is not easy the find the maximal size depending on p and k... Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

34 p = 2 p = 3 p = 4 k = S (k,p) 3 8,4 k = k = k = S (k,p) 8 8,4 k = k = k = S (k,p) 20 8,4 k = k = S (k,p) 60 8,4 k = k = Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

35 On the non-expandability The set S (k,p) m n,q can be expanded: fixed (α, β), let Consider the matrix B S (k+1,p) m n,q α α β α β β S m n,q(α, (k,p) β) A = β β α α α β β β S (k+1,p) m n,q (α, β) B = α α α β α β β β β β α α α α β β β β The last row of A can be overlapped with the first row of B. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35

36 Grazie per l attenzione. Antonio Bernini (DIMAI - Unifi) A 2D non-overlapping code Convegno GNCS / 35