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imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

Classification Relative to Hierarchical Order and Extension Property

Luciano Vianna F´ elix

Demat - UFRRJ

2015, January

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert

imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

Poset codes

Let Fq a finite field with q elements and V = Fn

q the

n-dimensional vector space of n-tuples over Fq. A [n, k]q linear code C is a k-dimensional subspace of Fn

q

If is a partial order defined over the elements of X, the pair P = (X, ) is a partially ordered set. A subset I ⊆ X is an ideal of P if the relation i ∈ I, j ≤ i implies that j ∈ I. Let A ∈ X, A denotes the ideal generated by A and is the smallest ideal of P that contains A. Let P = ([n], ) a poset and x ∈ Fn

q, we define the P-weight

• f x as ωP = |supp(x)| and the P-metric is

dP(x, y) = ωP(x − y).

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert

imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

Hierarchical Poset Suppose that [n] = {1, 2, . . . , n is a disjoint union [n] = H1 ∪ H1 ∪ · · · ∪ Hh with the relation given by i ≺ j if and

• nly if i ∈ Hs, j ∈ Ht and s < t. This order defines a hierarchical

poset on [n]. Canonical Decomposition of C.[2] Let C a [n, k]q hierarchical code. Then C is equivalent to a code C′ that can be decomposed as follows C = C1 ⊕ C2 ⊕ · · · ⊕ Ch Where supp (Ci) ⊆ Hi, Ci = 0 if i / ∈ Λ(C) and h

i=1 dim (Ci) = k.

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert

imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

Metric Invariants of Hierarchical Codes

Using the Canonical Decomposition of hierarchical codes, we can find some metric Invariants, such as: Minimal distance, the Generalized P-Weights, the Packing Radius. It allows us also to determine which hierarchical codes are MDS and Perfect and to describe a syndrome decoding algorithm with exponential gain when compared to the usual syndrome decoding of codes in a Hamming space and exponential loss when compared to a NRT poset code

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert

imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

In [3] we find a relation between the poset automorphism and the isometries in their poset spaces. We also study in which families of posets it is possible to extend an isomorphism of ideals to an automorphism of the poset. Generalizing the idea of [1] one can define the shape of an element

• f (Fn

q, dP), a function that classifies the orbit under isometries in

poset spaces. In [3] it is studied for what families of posets P one can define the shape function in (Fn

q, dP).

We have 4 standard operations over posets ([4]). Supposing that we start with posets P = ([n], P) and Q = ([m], Q) that have the IE property. We are interested if: a) Does P ∗ Q have the IE-property? b) In affirmative case, what are the shape of codevectors on P ∗ Q given the shapes on P and Q?

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert

imecc Structure and Classification of Hierarchical Posets IE Property and Classification of Orbits of Linear Isometries

Bibliography

• A. Barg and P. Purkayastha.

Bounds on ordered codes and orthogonal arrays. Moscow Math. J., 2:211–243, 2009.

• L. Felix; M. Firer.

Canonical-systematic form for codes in hierarchical poset metrics. Advances in Mathematics of Communications, 6:315–328, 2012.

• A. Barg; L. Felix; M. Firer; M. Spreafico.

Linear codes on posets with extension property. Discrete Mathematics, 317:1 – 13, 2014.

• R. P. Stanley.

Enumerative combinatorics. Cambridge University Press, Cambridg, 2012.

Luciano Vianna F´ elix Classification Relative to Hierarchical Order and Extension Propert