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Latin American Week on Coding and Information Covering problems in hierarchical poset spaces over finite rings

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos

INMA - UFMS and UEMS - Ponta Por˜ a

July - 2018

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem The Covering Problem

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem The Covering Problem Poset spaces

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem The Covering Problem Poset spaces Covering problems in poset spaces

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem The Covering Problem Poset spaces Covering problems in poset spaces Covering problem in hierarchical poset space

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Contents

The famous Football pool problem The Covering Problem Poset spaces Covering problems in poset spaces Covering problem in hierarchical poset space Short-covering problem in hierarchical poset space

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Football Pool

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Football Pool Problem

“ Which is the minimum number of bets necessary to guarantee n-1 correct results in n matches?´´

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Football Pool Problem

...in coding theory

Which is the minimum number of words in a code with the property that all words in the space Fn

3 are within Hamming

distance 1 from some codeword.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

The Covering Problem

Given integers n ≥ 1, q ≥ 2 and R ≥ 0, an alphabet A with |A| = q (An, d): the set of n-tuples with entries in A endowed with a metric d. B(c, R) = {x ∈ An : d(x, c) ≤ R}: the ball with center c ∈ An and radius R. Definition A subset C of An is a q-ary R-covering of An if

• c∈C

B(c, R) = An.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

The Covering Problem

Kq(n, R): the minimal size of a q-ary R-covering of lenght n.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

The Covering Problem

Kq(n, R): the minimal size of a q-ary R-covering of lenght n. The Covering Problem Is to determine Kq(n, R)

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Evolution

... for hamming distance and finite fields. 1948: Taussky and Todd introduce the numbers Kq(n, 1) from a group-theorethical viewpoint;

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Evolution

... for hamming distance and finite fields. 1948: Taussky and Todd introduce the numbers Kq(n, 1) from a group-theorethical viewpoint; 60’s: the problem received a lot of contributions and the problem was introduced in the coding theory context in the 60’s as covering codes with radius 1.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Evolution

... for hamming distance and finite fields. 1948: Taussky and Todd introduce the numbers Kq(n, 1) from a group-theorethical viewpoint; 60’s: the problem received a lot of contributions and the problem was introduced in the coding theory context in the 60’s as covering codes with radius 1. 80’s: initially investigated for arbitrary R by Carnielli.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Evolution

... for hamming distance and finite fields. 1948: Taussky and Todd introduce the numbers Kq(n, 1) from a group-theorethical viewpoint; 60’s: the problem received a lot of contributions and the problem was introduced in the coding theory context in the 60’s as covering codes with radius 1. 80’s: initially investigated for arbitrary R by Carnielli. Nowadays: Still an open problem.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Poset spaces

Poset P: Partially ordered set on {1, 2, ..., n}. Order Ideal: I ⊂ P is an ideal of P if a ∈ I, b ∈ P and b a then b ∈ I. the ideal generated by A: denote by A the smallest ideal containing A.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Rank of j ∈ P: l(j) = max{|C| : C ⊂ j and C is a chain}

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Examples of Posets

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Examples of Posets

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Examples of Posets

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Examples of Posets

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Poset Space

Xn

q: set of n-tuples with entries in a finite ring with q

elements. Support of a vector: supp(x) := {i ∈ P : xi = 0}. P-weight (ωP): ωP(x) := | supp(x) |. P-distance : dP(x, y) = ωP(x − y). Poset space: (Xn

q, dP).

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in poset spaces

Let An be an antichain on n. The metric dAn is the classical Hamming distance of coding theory. In 2008, Nakaoka and dos Santos introduced short-covering problem in Hamming spaces over finite rings: Given a integer R, which is the minimum number of words in a code with the property that all words in the space Xn

q are within

Hamming distance R from a multiple of some codeword.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in poset spaces

Let Cn be a chain poset. In 2010, Yildiz et al. solved the covering and short covering problems on this poset space over finite rings.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in poset spaces

Let n = mr and let n be a disjoint union m of chains of length

• r. The arising metric space is called the NRT space. In 2015,

Castoldi and Carmelo explore the covering problem in NRT spaces:

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in hierarchical poset spaces

From now, we will use Hn(n1, n2, ..., nh) to denote a hierarchical poset with h levels. A poset space defined by a hierarchical poset is called hierarchical poset space. K H

q (n(n1, n2, ..., nh); R): minimum size of a R-covering

code in the hierarchical poset space.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in hierarchical poset spaces

Denote by V H(n(n1, n2, ..., nh), R) the size(volumn) of a ball of radius R in the hierarchical poset space (Xn

q, dH). It is easy to

see that Theorem (Ball Covering Bound) K H

q (n(n1, n2, ..., nh); R) ≥

qn V H(n(n1, n2, ..., nh), R).

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in hierarchical poset spaces

Denote by V H(n(n1, n2, ..., nh), R) the size(volumn) of a ball of radius R in the hierarchical poset space (Xn

q, dH). It is easy to

see that Theorem (Ball Covering Bound) K H

q (n(n1, n2, ..., nh); R) ≥

qn V H(n(n1, n2, ..., nh), R). Theorem K H

q (n(n1, n2, ..., nh); R) ≤ qn−R.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering problems in hierarchical poset spaces

A minimal 1-covering code of (X3

2, H3(2, 1)) is

C = {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)}. So, K H

2 (3(2, 1); 1) = 4.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in hierarchical poset space

Let δi = i

j=1 nj and for a given integer R > 0 set a integer

0 < l ≤ h such that δl−1 < R ≤ δl and r = R − δl−1, where δ0 = 0. Note that, 1 ≤ ri ≤ ni.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in hierarchical poset space

Let δi = i

j=1 nj and for a given integer R > 0 set a integer

0 < l ≤ h such that δl−1 < R ≤ δl and r = R − δl−1, where δ0 = 0. Note that, 1 ≤ ri ≤ ni. Theorem K H

q (n(n1, n2, ..., nh); R) = Kq(nl, r)qn−δl.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in poset spaces

Let P and Q posets on n such that Q is a refinement of P, that is, P ⊂ Q. In this case, for x, y ∈ Xn

q, is easy to see

that dP(x, y) ≤ dQ(x, y).

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in poset spaces

Let P and Q posets on n such that Q is a refinement of P, that is, P ⊂ Q. In this case, for x, y ∈ Xn

q, is easy to see

that dP(x, y) ≤ dQ(x, y). Theorem K P

q (n; R) ≤ K Q q (n; R).

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in poset spaces

Let P and Q posets on n such that Q is a refinement of P, that is, P ⊂ Q. In this case, for x, y ∈ Xn

q, is easy to see

that dP(x, y) ≤ dQ(x, y). Theorem K P

q (n; R) ≤ K Q q (n; R).

Since, An ⊂ P ⊂ Cn for all poset P on n, we can derive the trivial upper bound for all poset metric space.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Covering in poset spaces

Let P and Q posets on n such that Q is a refinement of P, that is, P ⊂ Q. In this case, for x, y ∈ Xn

q, is easy to see

that dP(x, y) ≤ dQ(x, y). Theorem K P

q (n; R) ≤ K Q q (n; R).

Since, An ⊂ P ⊂ Cn for all poset P on n, we can derive the trivial upper bound for all poset metric space. Corollary Kq(n; R)

(1)

= K An

q (n; R) ≤ K P q (n; R) ≤ K Cn q (n; R) (2)

= qn−R.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Short-covering in hierarchical poset spaces

Ambient space: Xn

q;

metric: hierarchical Extended ball: E(c, R) =

• α∈Xq

B(αc, R) Definition Short-covering H ⊂ Xn

q is an R-short covering of a metric space

(Xn

q, d) if

• h∈H

E(h, R) = Xn

q.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Short-covering in hierarchical poset spaces

On the hierarchical poset space (X3

3, H3(2, 1)) an 1-short

covering code is given by C = {(0, 0, 1), (1, 1, 0), (1, 1, 1), (1, 1, 2)}. One can easily check that C is minimal, so CdH(X3, 3, 1) = 4.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Short-covering in hierarchical poset spaces

D(Xq)∗: zero-divisors of Xq, except the zero. U(Xq): the unity elements of Xq. Theorem Let H = Hn(n1, n2, ..., nh) be the hierarchical poset over n, with level distribution n1, n2, . . . , nh. For i = 1, ..., h, denote δi = i

j=1 nj and for a given integer R > 0 such that

δl−1 < R ≤ δl, holds (i) If R = δl, then CH(Xq, n, R) = C(Xq, n − R, 0). (ii) If δl−1 < R < δl, then CH(Xq, n, R) ≤ C(Xq, nl, r)

• qn−δl + s(s − 1)|U(Xq)|C(Xq, n − δl, 0)
• +

C(Xq, n − δl, 0), where r = R − δl−1 and s is the number of orbits of D(Xq)∗.

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings

Thank you!

Marcos V. P . Spreafico and Ot´ avio J. N. T. N. dos Santos Covering problems in hierarchical poset spaces over finite rings