Matched Metrics to the Binary Asymmetric Channels
11 Matched Metrics to the Binary AsymmetricChannels
Claudio M. Qureshi
Abstract
In this paper we establish some criteria to decide when a discrete memoryless channel admitsa metric in such a way that the maximum likelihood decoding coincides with the nearest neighbordecoding. In particular we prove a conjecture presented by M. Firer and J. L. Walker establishing thatevery binary asymmetric channel admits a matched metric.
Index Terms
Binary asymmetric channel, channel model, maximum likelihood decoding
I. I
NTRODUCTION
As it is well known, maximum likelihood decoding (MLD) over a symmetric channel coincideswith nearest neighbor decoding (NND) with respect to the Hamming metric. In this paper wedeal with the problem of matching a metric to a given channel regarding the decoding criteriamentioned above. This problem was considered in 1967 by J. L. Massey [2] where a metricmatched to a discrete memoryless channel is defined as a metric for which the NND is a MLD.Since then, this type of matching have been studied in some special cases. For instance, certainchannels matching to the Lee metric were obtained in [1]. The problem of matching a metric toa channel was taken up by G. Séguin [3], where the main focus was on sequences of additivemetrics. In the referred paper it is used a stronger condition also assumed here: a metric matchedto a channel is one for which not only the NND is a MLD but also the MLD is a NND. The
The author was supported by CNPq (grant 150270/2016-0) and FAPESP (grants 2015/26420-1 and 2013/25977-7) andis with the Institute of Mathematics, Statistics and Computing Science of the University of Campinas, SP , Brazil (email:[email protected]).Digital Object Identifier: 10.1109/TIT.2016.xxxxxxx
November 8, 2018 DRAFT a r X i v : . [ c s . I T ] M a r author obtains necessary and sufficient conditions for the existence of additive metrics matchedto a channel and raises the question of what happens if the restriction of additivity of the metricis removed. There was no significant progress until the paper [4] by M. Firer and J. Walker,where the authors proved, among other results, the existence of a metric (not necessarily additive)matched to the Z-channels and to the n -fold binary asymmetric channel (BAC) for n = , , andconjectured that this is also true for n > . Some recent progress in this direction was obtained in[5] where it is presented an algorithm to decide if a channel is metrizable and in that case returna metric matched to the channel, and in [6] where the author proved that the BAC channels aremetrizable in the weaker sense of J. L. Massey in [2].The main results of this paper are Theorem 1, which establishes a necessary and sufficientcondition for metrizability of a channel in terms of graph theory, and Theorem 2 which establishesthat the BAC channels are metrizable. Other contributions are the association of channels withgraphs (which allows the use of techniques from graph theory to approach problems related tochannels) and the introduction of a new structure: the colored posets, which may also be usefulin other contexts. This work is organized as follows: In Section II we give a brief review ofdefinitions and concepts needed in the development of the paper. In Section III we associate agraph with a channel and discuss some results of [4] and [6] in terms of this graph. In Section IVwe introduce the concept of colored poset which is used to describe an algorithm for constructinga metric matched to a channel whenever it is metrizable. A necessary and sufficient criterion formetrizability of a channel is also derived. In Section V this criterion is used to prove that theBAC channels are metrizable. In Section VI we introduce the concept of order of metrizabilityof a channel and settle some problems related to this.II. P RELIMINARIES
We summarize here some concepts and results to be used in the following sections.A discrete memoryless channel (simply referred as channel in this paper) W : X → X ischaracterized by its transition matrix related to the input and output alphabet X = { x , x , . . . , x N } .This matrix [ W ] ∈ M N × N ( R ) is given by [ W ] i j = Pr W ( x i | x j ) , the probability of receiving x i if x j was sent. When the channel is understood, this conditional probability is denoted by Pr ( x i | x j ) .Matrices associated with channels are characterized by the property that every entry is non-negative and the sum of the entries in each column is one. November 8, 2018 DRAFT
A channel W : X → X is metrizable (in the strong sense of [4] and [3]) if there is a metric d : X × X → [ , ∞) (i.e. d is a definite-positive symmetric function satisfying the triangleinequality) such that every nearest neighbor decoder is a maximum likelihood decoder and viceversa. This is also equivalent to each of the following statements:i) For all x ∈ X and every code C ⊆ X we havearg max y ∈ C Pr ( x | y ) = arg min y ∈ C d ( x , y ) , where both arg max and arg min are interpreted as returning lists of size at least .ii) For all x , y , z ∈ X the following condition holds:Pr ( x | y ) ≤ Pr ( x | z ) ⇔ d ( x , y ) ≥ d ( x , z ) . In this paper we only deal with reasonable channels (in the sense of [4]), that is, channels W : X → X such that Pr ( x | x ) > Pr ( x | y ) , ∀ x , y ∈ X with y (cid:44) x , (1)which is a necessary condition for a channel to be metrizable.Let W : X → X be a channel, X ( ) = { A ⊆ X : A = } be the family of -subsets of X and h : X ( ) → [ , + ∞) be a non-zero function. We say that h is coherent-with- W ifPr ( x | y ) ≤ Pr ( x | z ) ⇔ h ({ x , y }) ≤ h ({ x , z }) , for all x , y , z ∈ X with x (cid:44) y and x (cid:44) z . If such a function exists, we can construct a metricmatched to the channel as follows. Proposition 1.
Let W : X → X be a channel and h : X ( ) → [ , + ∞) be a coherent-with-Wfunction with maximum value m = max { h ( x ) : x ∈ X ( ) } . The function d : X × X → [ , + ∞) given by: d ( x , y ) = m − h ({ x , y }) if x (cid:44) y , if x = y , (2) is a metric matched to the channel W .Proof. To prove that d is positive-definite, we note that d ( x , y ) = m − h ({ x , y }) ≥ m > . Thefunction d is clearly symmetric since { x , y } = { y , x } . To prove triangle inequality, we consider x , y , z ∈ X pairwise distinct and note that d ( x , y ) + d ( y , z ) = m − h ({ x , y }) − h ({ y , z }) ≥ m ≥ m − h ({ x , z }) = d ( x , z ) . Therefore d is a metric. This metric matches to the channel W because November 8, 2018 DRAFT it is reasonable and for x , y , z ∈ X pairwise distinct we have Pr ( x | y ) ≤ Pr ( x | z ) ⇔ h ({ x , y }) ≤ h ({ x , z }) ⇔ d ( x , y ) ≥ d ( x , z ) , where the first equivalence is because h is coherent-with- W andthe second by the definition of d . (cid:3) The binary ( -fold) asymmetric channel with parameters ( p , q ) ∈ [ , ] (denoted by B AC ( p , q ) )is the channel with input and output alphabet Z = { , } and conditional probabilities Pr ( | ) = p and Pr ( | ) = q (and Pr ( | ) = − p and Pr ( | ) = − q ). The n -fold binary asymmetricchannel B AC n ( p , q ) is the channel with input and output alphabet X = Z n and for x = ( x , . . . , x n ) and y = ( y , . . . , y n ) in Z n the conditional probabilities are given byPr ( x | y ) = n (cid:214) i = Pr ( x i | y i ) . We remark that the channel
B AC n ( p , q ) verifies condition (1) if and only if p + q < (thereforeonly this case will be considered in this paper). Indeed, for n = it is obvious and for n > it is a direct consequence of Equation (4) in Section V. The metrizability of B AC n ( p , q ) wasestablished in [4] for the case pq = and n arbitrary (the n -fold Z -channel) and for p + q < and n = , . The remaining case is when p + q < and pq > . For this case, we prove thatthe corresponding channels are metrizable in Theorem 2.A partially ordered set (or poset) is a pair ( P , ≤) where ≤ is a partial order relation (i.e. itis reflexive, antisymmetric and transitive). The poset is denoted by P when the order relationis understood. Each poset is associated with a Hasse diagram, which is a representation of theposet in such a way that if x < y the element y is above x , and there is a segment connectingthese points whenever there is no z ∈ P with x < z < y . A directed graph (or digraph) G isdetermined by a pair ( V , E ) where V is a set, called the vertex set, and E ⊆ V × V is the edgeset. When ( v , w ) ∈ E we say that the edge v → w belongs to G . A path in G of length r ≥ isa finite sequence of vertices c = ( v , . . . , v r ) such that v i → v i + belongs to G for ≤ i < r . If c (cid:48) = ( w , . . . , w s ) is other path in G with w = v r , the path c ∗ c (cid:48) : = ( v , . . . , v r = w , w , . . . , w s ) isalso in G and it is called the concatenation of c and c (cid:48) . The reverse path of c is c = ( v r , . . . , v ) ,which is not necessarily a path in G . When r ≥ and v = v r , the path c = ( v , . . . , v r ) is calleda directed cycle. A digraph without directed cycles is called acyclic. From a directed acyclicgraph G = ( V , E ) we have a natural poset structure on V defining x ≤ y if there is a (directed)path (of length r ≥ ) from x to y . When we refer to the Hasse diagram of a directed acyclicgraph G we mean the Hasse diagram of their associated poset. November 8, 2018 DRAFT
III. T
HE GRAPH G ASSOCIATED WITH A CHANNEL
We associate with each channel (given by its transition matrix) a graph which plays animportant role in the proof of the metrization of the BAC channel.
Definition 1.
Let W : X → X be a channel. The digraph G ( W ) has vertex set X ( ) , the familyof -subsets of X , and directed edges linking { i , j } to { i , k } when Pr ( i | j ) < Pr ( i | k ) . Example 1.
Let W : X → X be a channel with X = { a , b , c , d } and transition matrix [ W ] = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) . . . . . . . . . . . . . . . . (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . Denoting by x y the set { x , y } , the vertex set of G ( W ) is X ( ) = { ab , ac , ad , bc , bd , cd } . Todetermine the edges we have to compare the conditional probabilities in each row of [ W ] (without taking into account the main diagonal). The first row gives us the following information:Pr ( a | c ) < Pr ( a | d ) < Pr ( a | b ) so, we obtain the following edges: ac → ad, ac → ab andad → ab. Looking at the second row we have no inequalities among Pr ( b | a ) , Pr ( b | c ) andPr ( b | d ) so, there are no new edges among the vertices ab , bc and bd. The third row givesus the inequalities: Pr ( c | b ) < Pr ( c | a ) and Pr ( c | d ) < Pr ( c | a ) which generate the followingnew edges bc → ac and cd → ac. Finally, from the fourth row we obtain the inequalities:Pr ( d | a ) < Pr ( d | b ) < Pr ( d | c ) from which we have the new edges ad → bd, ad → cd andbd → cd. Thus, the graph G ( W ) has edges and it is represented in Figure 1. Fig. 1. The digraph G ( W ) associated with the channel W of Example 1. November 8, 2018 DRAFT
A sufficient condition to guarantee the non-existence of a metric matched to a given channel W is given in Proposition 5 of [4]. This condition states that if the channel W : X → X admitsa decision chain of length r ≥ it is not metrizable. A decision chain of length r is a sequence x , x , . . . , x r − ∈ X verifying Pr ( x i | x i − ) < Pr ( x i | x i + ) for i : 0 ≤ i < r , where the indicesare taken modulo r (we note that the definition given in [4] in terms of t -decision region isequivalent to the one given here). Proposition 5 of the referred paper can be rewritten, in termsof the graph G ( W ) , as follows. Proposition 2.
If a channel W is metrizable, then its associated graph G ( W ) is acyclic. The following example shows that the converse is false.
Example 2.
Consider the channel W : X → X where X = { , , } with transition matrix [ W ] = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) / / / /
36 25 /
36 5 / /
12 1 /
18 7 / (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . In this case the graph G ( W ) has three vertices and only two edges: { , } → { , } and { , } → { , } therefore it is acyclic. However a metric compatible with W should verifyd ( , ) = d ( , ) = d ( , ) < d ( , ) = d ( , ) < d ( , ) = d ( , ) which is impossible, therefore Wis not metrizable. Definition 2.
Let W be a channel. The graph G ( W ) is transitive if for every path ( v , v . . . , v r − ) with ( v ∩ v r − ) = , the edge v → v r − belongs to G ( W ) . It is easy to see that if G ( W ) is transitive, then it is acyclic. The converse is false (the samechannel of Example 2 provides a counterexample). Proposition 2 can be strengthened as follows. Proposition 3.
If a channel W is metrizable, then its associated graph G ( W ) is transitive. This proposition has straightforward verification (it can also be obtained as a particular case ofTheorem 1 in Section V). Since every directed acyclic graph can be associated with a poset (inthe way mentioned at the end of Section II), we can associate a poset with a channel wheneverits associated graph is acyclic. The transitivity of the graph G ( W ) means that if v < w and v ∩ w (cid:44) ∅ , then v → w is and edge of G ( W ) . November 8, 2018 DRAFT
Let W : X → X be a channel. In terms of the conditional probabilities of W , the condition forthe graph G ( W ) to be transitive can be written as follows: G ( W ) is transitive if and only if everysequence x , x , . . . , x r − ∈ X ( r ≥ ) satisfying x i (cid:44) x i + , x (cid:44) x r − and Pr ( x i | x i − ) < Pr ( x i | x i + ) for ≤ i ≤ r − (indices taken modulo r ) also satisfies Pr ( x r − | x ) < Pr ( x r − | x r − ) . This isexactly the condition proposed in [6] to guarantee the existence of a metric d such thatarg max y ∈ C Pr W ( x | y ) ⊇ arg min y ∈ C d ( x , y ) , (3)for all C ⊆ X and x ∈ X (interpreting both arg max and arg min as returning list of size at least ). Using this condition the author also proves that the BAC channels admit a metric verifying(3). The reciprocal of Proposition 3 is also false (in other words it is not possible to proveequality in equation (3) under the hypothesis of transitivity). Example 3.
Let W : X → X be the channel with alphabet X = { , , , } and matrix transition [ W ] = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) .
44 0 .
22 0 .
22 0 . .
26 0 .
52 0 .
26 0 . .
12 0 .
08 0 .
16 0 . .
18 0 .
18 0 .
36 0 . (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . The graph G ( W ) is transitive. This graph and its Hasse diagram is showed in Figure 2. Everycompatible metric should verify d ( , ) > d ( , ) = d ( , ) = d ( , ) = d ( , ) = d ( , ) = d ( , ) which is impossible, then W is not metrizable. Fig. 2. The transitive graph G ( W ) of Example 3 (left) and its Hasse diagram (right). November 8, 2018 DRAFT
IV. C
OLORED POSETS AND A NECESSARY AND SUFFICIENT CONDITION FOR THE CHANNELTO BE METRIZABLE
Let P be a poset. A chain of length r ≥ in P is a finite sequence ( x , . . . , x r ) such that x i < x i + for ≤ i < r . The height function relates to each element of a poset P , the maximumpossible length of a chain ending in such element. In this paper we refer to this function asthe standard height of P and we call height function to any function h : P → N verifying h ( x ) < h ( y ) whenever x < y . A subset X ⊆ P is called horizontal (with respect to h ) whenthe restriction of h to X is constant. We say that a height is complete when its image is of theform [ k ] = { , , . . . , k − } for some k ∈ Z + . To each height function h we can associate aHasse diagram such that y is above x if and only if h ( y ) > h ( x ) . This association establishesa bijection between complete heights and Hasse diagrams (in the sense that we can recover theheight from its Hasse diagram).As remarked in the previous section, when the graph G ( W ) associated with a channel W : X → X is acyclic, we can associate with it a poset P = P ( W ) on the set X ( ) . The standardheight verifies h ({ x , y }) < h ({ x , z }) whenever Pr ( x | y ) < Pr ( x | z ) and from this, assuming thechannel is reasonable (i.e. it verifies (1)), we can construct a metric as in Proposition 1, verifying d ( x , y ) > d ( x , z ) whenever Pr ( x | y ) < Pr ( x | z ) . In particular this metric verifies (3) and will beweakly metrizable in the sense of [6]. But this metric does not necessarily will match with thechannel, since the poset structure of W (when G ( W ) is transitive) does not give informationabout when two -subsets { x , y } and { x , z } verify Pr ( x | y ) = Pr ( x | z ) or not, except when theyare connected in G ( W ) . We need a more general structure to manage also with these cases.Let P be a poset. A coloration for P is any function c : P → C ( C is a finite set) verifyingthat c ( x ) (cid:44) c ( y ) if x < y . A subset X ⊆ P is monochromatic (with respect to c ) if the restrictionof c to X is constant. In particular every monochromatic set is an antichain of P . A coloredcycle is a sequence ( x , x , . . . , x r ) in P verifying that x = x r and x i < x i + if c ( x i ) (cid:44) c ( x i + ) for ≤ i < r . Trivial examples of colored cycles are of the form ( x , x , . . . , x r ) with x = x r and c ( x ) = c ( x ) = · · · = c ( x r ) , we call these cycles monochromatic. Definition 3.
A colored poset is a pair ( P , c ) where P is a poset and c a coloration for P suchthat every colored cycle in P is monochromatic. November 8, 2018 DRAFT
Example 4.
Consider P = { , , , , , } with the divisibility relation (i.e. x ≤ y if x divides y ). Let c and c be the colorations for P given by c ( ) = c ( ) = ’blue’ , c ( ) = c ( ) = ’red’ , c ( ) = c ( ) = ’black’ and c ( ) = c ( ) = ’blue’ , c ( ) = c ( ) = ’red’ , c ( ) = c ( ) = ’black’. Then, ( P , c ) is a colored poset but ( P , c ) is not, because it contains thenon-monochromatic cycle ( , , , , ) (see Figure 3). Fig. 3. Two different colorations for the divisibility poset P = { , , , , , } ; the first corresponds to a colored poset (left)and the second one does not (right). Let P be a poset, c : P → C be a coloration for P and G P be the digraph associated with P (i.e. its vertex set is P and x → y belongs to G P if x < y ). We remark that a colored cycleis not necessarily a cycle in G P , however colored cycles have a nice interpretation in term ofgraphs. Namely, if G P ( c ) denotes the digraph obtained from G P adding the edges x → y with c ( x ) = c ( y ) , the colored cycles of P (with respect to c ) correspond to cycles of G P ( c ) .A Hasse diagram for a colored poset ( P , c ) is a Hasse diagram for P with the additionalproperty that if two points have the same color they are in the same level (i.e. no one is aboveor below the other). The next proposition guarantee the existence of a Hasse diagram for coloredposets. Proposition 4.
Let ( P , c ) be a colored poset. There exists a height function h for P such thatevery monochromatic subset of P is horizontal with respect to h.Proof. Let c : P → { c , . . . , c k } be the coloration, A i = c − ( c i ) for ≤ i ≤ k and P / c : = { A , . . . , A k } . We note that P / c is a partition of P into monochromatic set, with c ( A i ) (cid:44) c ( A j ) if i (cid:44) j and every monochromatic subset of P is contained in some A i . Thus, it suffices to constructa height function h for P such that every A i is horizontal. For X , Y ⊆ P we write X < Y if there November 8, 2018 DRAFT0 exist x ∈ X and y ∈ Y such that x < y . We claim that it is possible to order the indices of theelements of P / c in such a way that if A i < A j then i < j . Indeed, consider the digraph G whosevertex set is P / c and edges of the form A i → A j with A i < A j . This digraph is acyclic, becauseif there is a cycle c = ( A i , . . . , A i k ) in G with k ≥ , then for each j = , . . . , k − there are x i j ∈ A j and y i j + ∈ A j + with x i j < y i j + . Thus, the colored cycle ( x , y , x , y , . . . , x k − , y k , x ) is non-monochromatic (because x < y implies c ( x ) (cid:44) c ( y ) ) which is a contradiction since ( P , c ) is a colored poset. If ( P / c , (cid:52) ) is the poset induced by the acyclic digraph G (i.e. A i (cid:52) A j if there is a path from A i to A j in G ), clearly A i < A j implies A i (cid:52) A j . By extending this posetto a total order we have A i (cid:52) A i (cid:52) · · · (cid:52) A i k where i , i , . . . , i k is a permutation of , , . . . , k .Thus we can assume that if A i < A j then i < j (ordering indices if necessary).Next we define inductively an increasing sequence h , h , . . . , h κ of heights for P such that A j is horizontal with respect to h i if j ≤ i . We write x ≥ A for x ∈ P and A ∈ P / c when x ≥ a for some a ∈ A (otherwise we write x (cid:3) A ). We start considering the standard height h and t = max { h ( x ) : x ∈ A } . We define h as follows. h ( x ) = max { h ( x ) + t − h ( a ) : a ∈ A , a ≤ x } , if x ≥ A , h ( x ) , otherwise.We claim that this function is a height function for P and h ( A ) = { t } (in particular A ishorizontal with respect to h ). Indeed, let x , y ∈ P with x > y . We consider three cases: (i) x > y ≥ A , (ii) x ≥ A and y (cid:11) A and (iii) x , y (cid:11) A . In the first case, since for every a ∈ A with a ≤ y we have a ≤ x and h ( y ) + t − h ( a ) < h ( x ) + t − h ( a ) ≤ h ( x ) for all a ∈ A with a ≤ y , then h ( y ) < h ( x ) . In the second and third cases we have h ( x ) ≥ h ( x ) > h ( y ) = h ( y ) .In all the cases we conclude that h ( x ) > h ( y ) , thus h is a height for P . Moreover, if a ∈ A then a ≥ A , and since A is an antichain we have h ( a ) = h ( x ) + t − h ( a ) = t .Now we assume that there exists a height function h m for which A , . . . , A m are horizontal( ≤ m < κ ) and let t m + = max { h m ( x ) : x ∈ A m + } . We define: h m + ( x ) = max { h m ( x ) + t m + − h m ( a ) : a ∈ A m , a ≤ x } if x ≥ A m and h m + ( x ) = h m ( x ) otherwise. Using a similar argument to the case m = (considering three cases) we can prove that h m + is a height function (i.e h m + ( x ) > h m + ( y ) whenever x > y ). Since A m + is an antichain, then h m + ( a ) = t m + for all a ∈ A m + . Let x ∈ A i for some i , ≤ i ≤ m . We have x (cid:11) A m + because otherwise we would have A i ≥ A m + with i < m + which is a contradiction. Therefore h m + ( a ) = h m ( a ) which, by inductive hypothesis, November 8, 2018 DRAFT1 does not depend on a ∈ A i . In the last step (when m = k ) we obtain a height function h k forwhich all the elements of P / c are horizontal. In particular, since every monochromatic subset iscontained in some A i , every monochromatic subset is horizontal with respect to h k . (cid:3) Remark 1.
Since the proof of Proposition 4 is constructive, it brings us an algorithm to constructa Hasse diagram for a colored poset ( P , c ) . We start constructing the standard height functionfor P (first step) and after at most k steps we obtain a height function which induces a Hassediagram for ( P , c ) , where k is the number of colors. We say ’at most k steps’ instead of k stepsbecause when A i + is horizontal with respect to h i we have h i + = h i (this happens for examplewhen A i + has a unique element) and we can omit this step. Example 5.
Consider the colored poset ( P , c ) where P = Z × Z with the order induced by < < , < < and < ; and the coloration c : P → { R , B , G , D } given by (cid:101) R = { , } , (cid:101) B = { , } , (cid:101) G = { } and (cid:101) D = { } , where (cid:101) X : = c − ( X ) . The relation consideredat the beginning of the proof of Proposition 4 restricted to P / c is: (cid:101) R < (cid:101) G , (cid:101) R < (cid:101) B , (cid:101) G < (cid:101) Band (cid:101) D < (cid:101) R, which can be extended to the total order (cid:101) D (cid:52) (cid:101) R (cid:52) (cid:101) G (cid:52) (cid:101) B. Therefore definingA = (cid:101) D , A = (cid:101) R , A = (cid:101) G and A = (cid:101) B, we have that if A i < A j then i < j. Figure 4 shows thedifferent steps for the construction of a Hasse diagram for this colored poset. In the final stagewe obtain the height function h : P → N given by h ( ) = , h ( ) = h ( ) = , h ( ) = and h ( ) = h ( ) = . Fig. 4. The construction of a Hasse diagram for a colored poset.
Our next goal is to associate with each channel W (under certain conditions) a colored posetand to construct a metric from a height function for its Hasse diagram. We start by introducingthe graphs G( W ) and G ( W ) associated with the channel W . November 8, 2018 DRAFT2
Definition 4.
Let W : X → X be a channel and X ( ) be the family of 2-subsets of X . Thedigraph G( W ) has vertex set X ( ) and directed edges linking { x , y } to { x , z } if y (cid:44) z andPr ( x | y ) ≤ Pr ( x | z ) . The graph G ( W ) is a non-directed graph whose vertex set is X ( ) and twovertices { x , y } and { x , z } are connected by an edge in G ( W ) if y (cid:44) z and Pr ( x | y ) = Pr ( x | z ) . The graph G ( W ) can be identified with the subgraph of G( W ) whose vertex set is X ( ) andedges ν → ω and ω → ν for each edge { ν, ω } in G ( W ) . By construction the graphs G ( W ) and G ( W ) have no common edges. We use this identification in this paper. Definition 5.
A digraph G is cycle-reverter if for each cycle c = ( v , v , . . . , v r − , v ) in G, thenthe reverse cycle c = ( v , v r − , . . . , v , v ) is also in G We remark that if G( W ) is cycle-reverter then the graph G ( W ) is transitive (the converse isfalse) and, in particular, acyclic. Lemma 1.
Let W : X → X be a channel such that its associated graph G( W ) is cycle-reverterand let A = { A , . . . , A k } be the set of connected components of G ( W ) . Consider for P : = X ( ) the poset structure induced by the graph G ( W ) and the function c : P → A given by c ( v ) = Aif v ∈ A. Then ( P , c ) is a colored poset.Proof. First we prove that c is a coloration for P . Let v , w ∈ P such that v < w , then thereexists a path p = ( v = v , v , . . . , v r = w ) (with r ≥ ) in G ( W ) . In particular v = { x , y } and v = { x , z } with Pr ( x | y ) < Pr ( x | z ) . We suppose, to the contrary, that c ( v ) = c ( w ) . Since v and w belong to the same connected component in G ( W ) there is a path p from w to v in G ( W ) .Since G( W ) is cycle-reverter, the reverse of the cycle p ∗ p is also in G( W ) . In particularthe arrow v → v is in G( W ) and then Pr ( x | z ) ≥ Pr ( y | z ) , which is a contradiction. Now weprove that ( P , c ) is a colored poset. Consider a colored cycle C = ( v , v , . . . , v r ) with v = v r . If v i < v i + there is a path from v i to v i + in G ( W ) and if c ( v i ) = c ( v i + ) there is a path from v i to v i + in G ( W ) . In both cases there is a path p i from v i to v i + in G( W ) for ≤ i < r . Sincethe graph G( W ) is cycle-reverter the reverse of the cycle p ∗ p ∗ . . . ∗ p r − is a cycle in G( W ) .Therefore none of the paths p i can be in G ( W ) and we conclude that all the vertices v i belongto the same connected component in G ( W ) , then C is monochromatic. (cid:3) The next theorem establishes a necessary and sufficient condition for the existence of a metricmatched to a given channel.
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Theorem 1.
Let W : X → X be a channel. The graph G( W ) is cycle-reverter if and only if thechannel W is metrizable.Proof. First we suppose the existence of a metric d : X × X → [ , + ∞) matched to W andconsider a cycle c = ( v , v , . . . , v r = v ) in G( W ) . Every vertex is of the form v i = { x i , y i } with x i , y i ∈ X and ( v i ∩ v i + ) = for ≤ i < r (indices taken modulo r ). Since d matches with W ,from the cycle c , we obtain the following chain of inequalities: d ( x , y ) ≤ d ( x , y ) ≤ · · · ≤ d ( x r − , y r − ) ≤ d ( x , y ) . Therefore every inequality is actually an equality and the reverse cycle c is also in G( W ) .This proves that the graph G( W ) is cycle-reverter whenever W is metrizable. Conversely, if G( W ) is cycle-reverter then by Lemma 1 we can define in P = X ( ) a colored poset structurewhere the order is induced by the graph G ( W ) and the connected components of G ( W ) aremonochromatic. By Proposition 4 we can construct a height function h for P such that everyconnected component of G ( W ) is horizontal. In particular Pr ( x | y ) < Pr ( x | z ) if and only if h ({ x , y }) < h ({ x , z }) . Hence, the function ( x , y ) (cid:55)→ h ({ x , y }) is coherent-with- W , then W ismetrizable (a metric can be constructed as in Equation (2)). (cid:3) Remark 2.
When G( W ) is cycle-reverter we have the following algorithm to obtain a metric dmatching to the channel W : X → X . Consider the colored poset in P = X ( ) whose partial order is induced by G ( W ) and thecoloring is given by the connected components of G ( W ) . Construct a height function h for P as in the proof of Proposition 4 (see also Remark 1)for which every monochromatic set is horizontal. Let m be the maximum value of h. By Proposition 1, a metric matched to W is given byd ( x , y ) = m − h ({ x , y }) if x (cid:44) y if x = y Example 6.
Let X = { , , , } and W : X → X be the channel with transition matrix: [ W ] = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) / /
90 2 / / / / / / / / / (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . November 8, 2018 DRAFT4
We consider for X ( ) the order induced by G ( W ) and the coloration c as in Lemma 1 (i.e. eachcolor correspond to a connected component of G ( W ) ). Figure 5 shows the graph G( W ) where theedges corresponding to the subgraphs G ( W ) and G ( W ) are colored black and red respectively.Colored cycles correspond to cycles in G( W ) . Note that a cycle and its reverse are in G( W ) if andonly if it is a cycle in G ( W ) . Since there are no cycles in G( W ) containing black edges, the graph G( W ) is cycle-reverter. Thus, by Lemma 1, (X ( ) , c ) is a colored poset. We can apply the stepsgiven in the proof of Proposition 4 to obtain a Hasse diagram for this colored poset. This processis illustrated in Figure 5, after three steps we obtain the height function h : X ( ) → N givenby h ({ , }) = , h ({ , }) = h ({ , }) = and h ({ , }) = h ({ , }) = h ({ , }) = . Ametric matched to W is given by d ( x , y ) = − h ({ x , y }) when x (cid:44) y and otherwise. Fig. 5. The graph G( W ) for the channel of Example 6 and the steps to obtain a metric matched to this channel. V. T HE BAC
CHANNEL IS METRIZABLE
We consider the n -fold BAC channel B AC n ( p , q ) with parameters p , q ∈ [ , ] and p + q < .The case pq = corresponds to the Z -channels which we know they are metrizable (Theorem 6of [4]), then we can assume pq > . Each entry of its transition matrix M n ( p , q ) is of the formPr ( x | y ) = p a ( − p ) b q c ( − q ) d November 8, 2018 DRAFT5 with a + b + c + d = n and a + d = w ( x ) (the Hamming weight of x ∈ Z n ). If we consider otherword y (cid:48) ∈ Z n with Pr ( x | y (cid:48) ) = p a (cid:48) ( − p ) b (cid:48) q c (cid:48) ( − q ) d (cid:48) , since a + d = a (cid:48) + d (cid:48) and b + c = b (cid:48) + c (cid:48) ,taking the quotient we have: Pr ( x | y ) Pr ( x | y (cid:48) ) = (cid:18) − pq (cid:19) b − b (cid:48) · (cid:18) − qp (cid:19) d − d (cid:48) . (4)This identity will be useful in our proof of metrizability of the BAC channel. Lemma 2.
Let W : X → X be a channel. The graph G( W ) is cycle-reverter if and only ifevery sequence x , x , . . . , x r − ∈ X (r ≥ ) satisfying x i (cid:44) x i + and Pr ( x i | x i − ) ≤ Pr ( x i | x i + ) for i : 0 ≤ i < r also satisfy Pr ( x i | x i − ) = Pr ( x i | x i + ) for i : 0 ≤ i < r (where the indices areconsidered modulo r).Proof. We suppose that G( W ) is cycle-reverter and consider a sequence x , x , . . . , x r − ∈ X satisfying x i (cid:44) x i + and Pr ( x i | x i − ) ≤ Pr ( x i | x i + ) for i : 0 ≤ i < r . Then we have a cycle c = ( v , . . . , v r − , v r = v ) in G( W ) given by v i = { x i , x i + } for ≤ i < r . Since this graph iscycle-reverter its reverse cycle c is also a cycle in G( W ) which implies Pr ( x i | x i − ) = Pr ( x i | x i + ) for i : 0 ≤ i < r . Now we suppose that the graph G( W ) is not cycle-reverter. In this case we can finda sequence x , x , . . . , x r − ∈ X satisfying x i (cid:44) x i + and Pr ( x i | x i − ) ≤ Pr ( x i | x i + ) for i : 0 ≤ i < r where at least one inequality is strict. Indeed, consider a cycle c = ( v , v , . . . , v r = v ) in G( W ) of minimal length r ≥ whose reverse cycle c is not in this graph. Since c = c for cyclesof length r ≤ , we have r ≥ . We remark that the fact that its reverse cycle c is not in G( W ) is equivalent to the existence of some arrow in c which is also an arrow in G ( W ) . Thevertices in c are pairwise disjoint since otherwise we could take a sub-cycle of c containingsome arrow of G ( W ) contradicting the minimality of r . If for some i : 0 ≤ i < r we have that v i ∩ v i + ∩ v i + = { x } , then there exists y , z , t pairwise distinct such that v i = { x , y } , v i + = { x , z } and v i + = { x , t } . Thus Pr ( x | y ) ≤ Pr ( x | z ) ≤ Pr ( x | t ) and the arrow v i → v i + is also in G( W ) . Ifsome of the arrows v i → v i + or v i + → v i + is in G ( W ) then v i → v i + is also in G ( W ) , sowe could substituting these two arrows for the last obtaining a new cycle, whose reverse is notin G( W ) and length r − which contradict the minimality of r . Therefore v i ∩ v i + ∩ v i + = ∅ forall i : 0 ≤ i < r and there exists a sequence x , x , . . . , x r − ∈ X such that v i = { x i , x i + } . Thissequence satisfies Pr ( x i | x i − ) ≤ Pr ( x i | x i + ) for i : 0 ≤ i < r with at least one strict inequality(the corresponding to the edge in G ( W ) ). (cid:3) November 8, 2018 DRAFT6
Theorem 2.
Let n ≥ and ( p , q ) ∈ ( , ] with p + q < . Then, the channel W = B AC n ( p , q ) ismetrizable.Proof. By Theorem 1, it is enough to prove that its associated graph G( W ) is cycle-reverter.We assume, to the contrary, that this graph is not cycle-reverter and by Lemma 2 there exists asequence x , x , . . . , x r − ∈ X such that x i (cid:44) x i + andPr ( x i | x i − ) ≤ Pr ( x i | x i + ) , ∀ i : 0 ≤ i < r , (5)where the indices are taken modulo r and where at least one of these inequality is strict. Wewrite these conditional probability as Pr ( x i | x i − ) = p a i ( − p ) b i q c i ( − q ) d i for i : 0 ≤ i < r .Therefore Pr ( x i − | x i ) = p c i ( − p ) b i q a i ( − q ) d i for i : 0 ≤ i < r and applying Equation 4 weobtain Pr ( x i | x i + ) Pr ( x i | x i − ) = p c i + ( − p ) b i + q a i + ( − q ) d i + p a i ( − p ) b i q c i ( − q ) d i = (cid:18) − pq (cid:19) b i + − b i (cid:18) − qp (cid:19) d i + − d i , (6)for ≤ i < r . Multiplying these r inequalities we have r − (cid:214) i = Pr ( x i | x i + ) Pr ( x i | x i − ) = r − (cid:214) i = (cid:18) − pq (cid:19) b i + − b i (cid:18) − qp (cid:19) d i + − d i = (cid:18) − pq (cid:19) Σ r − i = ( b i + − b i ) (cid:18) − qp (cid:19) Σ r − i = ( d i + − d i ) = because b r = b , d r = d and Σ r − i = ( b i + − b i ) = Σ r − i = ( d i + − d i ) = . But by (5) this product isgreater than (since at least one inequality is strict) which is a contradiction. Therefore W ismetrizable. (cid:3) VI. C
ONCLUDING REMARKS AND FURTHER PROBLEMS
In this work we approach the problem of metrization for the n -fold BAC channels in thesense of the definition used in [3] and [4]. An existence proof and an algorithm to construct ametric matching to the BAC channels are provided. An interesting problem is to describe the set D ( n , p , q ) of metrics matching to the channel B AC n ( p , q ) . This set is non-empty by Theorem2 and closed under linear combinations with positive real coefficients, so we could look for aminimal generator for this set. Describing this set allows to choose good metrics according to agiven criterion. One possible criterion could be to select the metric according to how easy is to November 8, 2018 DRAFT7 compute it. Other possible criterion could be to select the metric according to how good it fitsthe channel in the sense of the next definition.
Definition 6.
Let W : X → X be a channel and d be a metric compatible to W . The metric d ismatched to the channel W with order n if d m is a metric matched to W m for all m : 1 ≤ m ≤ n,where d m : X m → X m is given byd m ( x , y ) = m (cid:213) k = d ( x i , y i ) and Pr W m ( x | y ) = m (cid:214) k = Pr W ( x i | y i ) . If d is matched to W with order n for all n ≥ , we say that the metric d matches completelyto W . We also define the order of metrizability of W as the maximum n (possibly infinite) for whichthere exists a metric matched to W with order n . It would be interesting to determine the orderof metrizability for the BAC channels or at least to determine which of these channels admit amatched metric with order n ≥ . In [3] it was approached the problem of determining when achannel W : X → X is completely metrizable for alphabets of lengths X = , and was provedthat the channel B AC ( p , q ) has a matched metric with order ∞ if and only if p = q (symmetricchannel). In this case the Hamming metric matches completely to the channel. ACKNOWLEDGEMENTS
The author would like to thank the anonymous reviewers for their valuable comments andsuggestions, Sueli Costa for her support and suggestions, and CNPq and FAPESP for theirsupport. R
EFERENCES [1] J. C. Y. Chiang, J. K. Wolf, “On channels and codes for the Lee metric”,
Inf. Control , vol. 19, no. 2, pp. 159-173, 1971.[2] J. L. Massey, “Notes on coding theory”,
Cambridge, MA, USA: MIT Press , 1967.[3] G. Séguin, “On metrics matched to the discrete memoryless channel”,
J. Franklin Inst. , vol. 309, no. 3, pp. 179-189, 1980.[4] M. Firer, J. L. Walker, “Matched metrics and channels”,
IEEE Transactions on Information Theory , vol. 62, no. 3, pp.1150-1156, 2016.[5] R. G. L. D’Oliveira, M. Firer, “Channel metrization”, preprint, arXiv:1510.03104 , 2016.[6] A. Poplawsky, “On matched metric and channel problem”, preprint arXiv:1606.02763v1 , 2016., 2016.