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Metrics that respect support

Roberto Assis Machado

University of CampinasUniversity of Illinois at Urbana-Champaign

[email protected]

July 26th 2018

Roberto A. Machado (Unicamp) Metrics that respect support July 26th 2018 1 / 25

Summary

1 Introduction

2 Conditional Sums of known weights

3 Representation of weights by directed graphs


Distance & Coding

Distances are in the realm of coding: 1

Figura: Huffman & Pless Fundamentals of Error-Correcting Codes


Distance & Coding


Figura: MacWilliams & Sloane Theory of Error-Correcting Codes , page 6.


Distance & Coding


Figura: Ling & Xing Coding Theory, a First Course .


The Hamming Metric

The big star

Given u, v ∈ Fnq, with u = (u1, u2, . . . , un), v = (v1, v2, . . . , vn)

dH(u, v) = |{i : ui 6= vi}|.


What makes the Hamming distance so special

It fits a Symmetric Channel

It matches the Binary Symmetric Channel:

dH(u, v) ≤ dH(u,w)

if and only if

P(v = received|u = sent) ≥ P(w = received|u = sent).

This implies that, for every given code C ⊆ Fnq, decoding using maximal

likelihood or minimal distance leads to the same result:

arg maxv∈C

Pr(u = received | v = sent) = arg minv∈C

d(u, v). (1)


Invariant metrics

Such matching allowed to explore the code properties as invariant metrics:

packing radius;

covering radius;

MacWilliams’ Identity;

equivalence of codes;

and so forth.


Metrics introduced in Coding Theory

Through the years, many metrics were introduced in the context of codingtheory.

Poset metrics of Brualdi [2];

Gabidulin’s combinatorial metrics [6];

Poset-block metrics [1];

Digraph metrics [5].

Shared properties

P1: Weight condition, i.e., d(u, v) = wt(u − v).

P2: Support condition, i.e., wtH(u) ≤ wtH(v), if supp(u) ⊂ supp(v). Inthis case, we say that the metric respects support.


Weight function

A map wt : Fnq → Z is a weight function if given u, v ∈ Fn

q:

wt(u) ≥ 0, and wt(u) = 0 ⇐⇒ u = 0.

supp(u) ⊂ supp(v)⇒ wt(u) ≤ wt(v), where

supp(u) = {i ∈ {1, 2, . . . , n} : ui 6= 0},

in this case we say that the weight function respects support.

A weight function wt determines a semi-metric by definingd(u, v) = wt(u − v).


Example

00 10

01 11

00 10

01 11


Possible decoding criteria in F22

Criterion Hamming Poset Poset-block Combinatorial

wt(10) = wt(01) < wt(11) X X X Xwt(10) = wt(01) = wt(11) X Xwt(10) < wt(01) = wt(11) X Xwt(10) < wt(01) < wt(11)

The decoding criteria 0 = wt(00) < wt(10) < wt(01) < wt(11) is notcovered by any of such known metrics.


How to explore weight functions

In order to understand the space of all metrics satisfying conditions P1 andP2, it is enough to study the space of all weight functions that respectsupport.There are two approaches to explore such a family:

By conditional sums of known metrics;

By representing each weight function as an edge-weighted directedgraph.


Conditional Sums of known weights

Linear isometries

Let wt1 and wt2 be weights respecting support. Then,

1 (Direct sum:) wt1 + wt2 respects support;

2 (k-sum:) wt1 ⊕k wt2 respects support, where

(wt1 ⊕k wt2)(u) =

{wt1(u), if wt1(u) < k ,

wt1(u) + wt2(u), if wt1(u) ≥ k .


Conditional sum reaches new decoding criteria

Criterion Hamming Poset Poset-block Combinatorial

wt(10) = wt(01) < wt(11) X X X Xwt(10) = wt(01) = wt(11) X Xwt(10) < wt(01) = wt(11) X Xwt(10) < wt(01) < wt(11)

The decoding criteria 0 = wt(00) < wt(10) < wt(01) < wt(11) can bereached by

wt(x) = wt2(x)⊕2 wtH(x) =

{wt2(x), if |supp(x)| < 2,

wtH(x) + wt2(x), if |supp(x)| = 2.,

where wt2(00) = 0, wt2(10) = 1 and wt2(01) = wt2(11) = 2 (a posetmetric).


Conditional sums of known weights

In a more general setting, the k condition can be replaced by anycondition C respecting support.

wt(x) = wt1(x)⊕C wt2(x) =

{wt1(x) + wt2(x), if C ,

wt1(x), otherwise.

C respecting support

A condition C respects support if supp(u) ⊂ supp(v) and u satisfies acondition C , then v also does.


Conditional sums of known weights

Classes of equivalences for conditional sums

Let wt be a weight function. Then, wt⊕C wt and wt are equivalent if,and only if, ⊕C

∼= ⊕k , for some k ∈ N.


Representation of weights by directed graphs

Consider an edge-weighted directed graph G (V ,E , δ), where

V = Fn2;

E = {(u, v) : supp(u) ⊂ supp(v) and dH(u, v) = 1};δ : E → Z.

So that, wt(w) =∑

(u,v)∈T δ((u, v)), where T is a trail from the nullvector to w .

Hamming weight

10 01

00

11

1 1

11


Every weight function admits a representation

Given a weight wt : Fnq → N, δ is defined as by δ((u, v)) = wt(v)− wt(u)

for every v = u + ei ∈ Fnq.

10 01

00

11

wt(10)

wt(11) − wt(01)

wt(01)

wt(11) − wt(10)


Constraints on δ to construct a weight function

To ensure that a δ-digraph induces a weight it is required some extraconstraints on δ.

10 01

00

11

1 0

11

δ((0, ei ) > 0;

The summation wt(w) =∑

(u,v)∈T δ((u, v)) can not depend on thetrail.


Classification of Gabidulin’s combinatorial metrics

Proposition

Let G (Fn2,E , δ) be a δ-digraph determined by a weight function wt. Then,

wt is combinatorial if, and only if, δ((u, v)) ∈ {0, 1}.


Linear Isometries

Induced by permutations

Let φ ∈ Sn be a permutation.fφ(x1, . . . , xn) = (xφ(1), . . . , xφ(n)) is a linear isometry if, and only if, fφ is aδ-digraph automorphism.

10 01

00

11

1 1

11

10 01

00

11

1 2

12

Permutation is linear isometry Permutation is not linear isometry


Linear Isometries

Increasing the support

Consider a map f :Fnq→Fn

q defined as f (ei ) =∑n

j=1 λjej , with λi 6= 0. If∑(x ,y)∈T δ((x , y)) = 0 for every trail from u ∈ Fn

2 with ui 6= 0 tou + f (ei )− ei , then f is a linear isometry.

0

e1 e2 e3

e1 + e2e1 + e3

e2 + e3

e1 + e2 + e3

1 1 1

0 0 0101

10

1


References

Marcelo Muniz Silva Alves, Luciano Panek, Marcelo Firer, et al.

Error-block codes and poset metrics.Advances in Mathematics of Communications, 2008.

Richard A. Brualdi, Janine Smolin Graves, and K.Mark Lawrence.

Codes with a poset metric.Discrete Mathematics, 147(1 - 3):57 – 72, 1995.

Michel Marie Deza and Elena Deza.

Encyclopedia of distances.In Encyclopedia of Distances, pages 1–583. Springer, 2009.

Rafael Gregorio Lucas D’Oliveira and Marcelo Firer.

Channel metrization.CoRR, abs/1510.03104, 2015.

T. Etzion, M. Firer, and R. A. Machado.

Metrics based on finite directed graphs and coding invariants.IEEE Transactions on Information Theory, PP(99):1–1, 2017.

EM Gabidulin.

Combinatorial metrics in coding theory.In 2nd International Symposium on Information Theory. Akademiai Kiado, 1973.

EM Gabidulin.

A brief survey of metrics in coding theory.Mathematics of Distances and Applications, 66, 2012.


Gabidulin’s Combinatorial metric

Basic settings, definitions and notation

Pn := {A : A ⊂ {1, 2, . . . , n}};Given X ⊂ {1, 2, . . . , n}, a family F ⊂ Pn is a covering of X ifX ⊂ ∪A∈FA;

We assume F is a covering of {1, 2, . . . , n}.

Definition: Combinatorial weight

Given u = (u1, . . . , un) ∈ Fnq, the F-combinatorial weight is defined by

wtF (x) = min{|A| : A ⊂ F and A is a covering of supp(x)}.


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