aa r X i v : . [ m a t h . C O ] F e b Toric Codes from Order Polytopes
Mahir Bilen Can and Takayuki Hibi Tulane University, New Orleans, Louisiana, US, [email protected] Department of Pure and Applied Mathematics, Graduate School of Information Science andTechnology, Osaka University, Suita, Osaka 565–0871, Japan [email protected]
February 9, 2021
Abstract
In this article we investigate a class of linear error correcting codes in relation withthe order polytopes. In particular we consider the order polytopes of tree posets andbipartite posets. We calculate the parameters of the associated toric variety codes.
Keywords: Toric code, parameters, poset polytope, order polytopes, shrubs,bipartite posetsMSC: 11T71, 06A07
In the present article we are concerned with a special class of algebraic-geometric codes [15]that are defined on toric varieties. Building on a work of S. Hansen [5], J. Hansen initiatedthe study of toric codes on polygons in [4]. This development quickly led to numerous newresults on the algebraic-geometric codes that are constructed on higher dimensional toricvarieties. The articles [10, 11, 12, 13] amplified the importance of combinatorial approach indetermining the parameters of the toric codes. Our goal in this article is to show that, theset of order polytopes form an interesting ground for the applications of such work.Let P be a poset whose elements are listed as ε , . . . , ε m . Let N denote the free Z -moduleon P , N := L mi =1 Z ε i . Let M denote the dual of N , that is M := Hom Z ( N, Z ). As before,the dual of the element ε i ( i ∈ { , . . . , m } ) in M will be denoted by e i . Let 2 P denote theset of all subsets of P . We define the function ρ : 2 P → N ⊗ Z Q by W P ε i ∈ W ε i . The order polytope of P , denoted by O P , is the convex hull the finite set { ρ ( W ) : W is an upper order ideal of P } . The face lattice of the polytope O P was first described by Geissinger [3], whose resultswere amplified by Stanley in [14]. A concrete description of the edges of O P can be found1n [8]. Following [6], we now introduce a class of toric varieties that are closely related to theorder polytopes. The set of all order ideals of P , denoted by J ( P ), is a distributive latticewith respect to inclusion. In particular, we have the joins (denoted by ∨ ) and the meets(denoted by ∧ ) of the elements of J ( P ). Let Y := { y α : α ∈ J ( P ) } be a set of algebraicallyindependent variables indexed by the order ideals. Then the Hibi toric scheme associatedwith P is the projective scheme Proj k [ Y ] /I , where I is the homogeneous ideal I = ( y α y β − y α ∧ β y α ∨ β : y α , y β ∈ Y ) . It turns out that the fan of X P is the normal fan of the order polytope O P .The purpose of our article is to investigate the parameters of the toric code of the definingpolytope O P of X P . The parameters that we speak of are called the “length,” the “dimen-sion,” and the “minimum distance.” Although our method applies to all finite posets, in thisarticle we focus on the minimum distance computation for the order polytopes of the rootedtrees only. Let P = { ε , . . . , ε m } be a rooted tree, where ε is the root. Our first main result(recorded as Theorem 4.4) states that minimum distance of the toric code C O P over a finitefield F q , where q >
3, is given by d ( C O P ) = ( q − a ( q − b , for some a and b such that a + b = m . In fact, we know precisely what a and b are.Let P be a polytope. The length of the associated toric code C P over F q is given by( q − dim P , where dim P is the dimension of the affine hull of P . Hence, in our case, thelength is given by ( q − dim O P = ( q − m , where m is the cardinality of the poset P . Onthe other hand, the dimension of a toric code of P is given by the number of lattice pointsin P . Therefore, in our case, it is given by the number of (upper) order ideals of P . For arooted tree with m vertices, this number (dimension) varies in the range m + 1 , . . . , m − ;it is equal to the number of order preserving maps σ : P → { , } .Let Q be a graded poset with 2 m elements ( m ∈ Z + ). If Q has m minimum elements,then we will call Q an ( m, m ) -bipartite poset . The second infinite family of toric codes thatwe consider come from the order polytopes of ( m, m )-bipartite posets. Our second mainresult (recorded as Theorem 5.4) states that the minimum distance of the toric code C Q overa finite field F q where q > d ( C O P ) = ( q − m ( q − m . The dimension of such a code varies in the range 2 m +1 − , . . . , m .Before closing this introduction, we want to mention a fact we inferred from our cal-culations. In general, a preferable linear error correcting code is the one that has a ratioof dimension / length while the ratio minimum distance / length is as large as possible. It isnatural to wonder if it is possible to increase these ratios for a toric code by switching to thepolar polytope. In this article we pay a close attention to the polar of the order polytope ofa graded poset. It turns out that, by a result of Hibi and Higashitani [7], the polar polytopeof a suitable dilation of O P , called the poset polytope of P , is reflexive and terminal. (We will2xplain these notions in the sequel.) These properties essentially imply that the number oflattice points of a poset polytope is much smaller compared to the number of lattice pointsof the order polytope. Hence, as far as the parameters of linear codes are concerned, theorder polytopes are better than the poset polytopes.The structure of our paper is as follows. In the next section we introduce our basicnotation regarding posets, polytopes, and toric codes. In the same section we briefly reviewsome results of Soprunov and Soprunova also. The purpose of Section 3 is to compare thestructures of the order polytopes and poset polytopes. We prove our first main result aboutthe toric codes defined by the rooted tree posets in Section 4. We prove our second mainresult about the toric codes defined by the ( m, m )-bipartite graphs in Section 5. In addition,in this section, we observe that (Lemma 5.1) the free sum of two order polytopes, O P ⊕ O Q ,is equivalent to the order polytope O P ⊕ Q , where P ⊕ Q stands for the ordinal sum of P and Q . Here, the equivalence is defined by the change of coordinates. In this article, by a poset we will always mean a finite poset. A lower order ideal in P is asubposet I such that for every y ∈ I , if x ≤ y in P , then x ∈ I . An upper order ideal in P is defined similarly where we replace the condition x ≤ y with y ≤ x .The set of all lower order ideals of P is denoted by J ( P ). This is a distributive latticewith respect to inclusion. The set of all upper order ideals also form of a distributive lattice,which is isomorphic to J ( P opp ), where P opp denotes the opposite poset to P . An orderreversing bijection between two posets will be called an anti-isomorphism . If P and Q aretwo isomorphic (resp. anti-isomorphic) posets, then we will write P ∼ = Q (resp. P ∼ = a Q ).Let x and y be two elements from P . If x ≤ y , and z ≤ z ≤ y implies that z = x or z = y , then y is said to cover x . Customarily, the cover relation is denoted by x ⋖ y .An antichain is a poset whose elements are all incomparable. The greatest possible sizeof an antichain in P is called the width of P . Dilworth’s theorem [2] states that the width isequal to the minimal number of chains that cover the set.A chain is a poset C := { x , . . . , x n } whose elements are linearly ordered, x (cid:12) x (cid:12) · · · (cid:12) x n . A maximal chain in a poset P is a chain C ⊆ P such that C is not a subposet ofany other chain in P . If C = { x , . . . , x k } is a chain, then the length of C is defined as k − P is called a graded (or ranked) poset if every maximal chain in P has the samelength. In this case, a function ℓ : P → Z which has the property that ℓ ( y ) = ℓ ( x ) + 1 forevery cover relation x ⋖ y in P is called a rank function for P . Without loss of generality weassume that ℓ ( x ) = 0 whenever x is a minimal element. Then ℓ is uniquely determined by P , so, we call it the rank function of P .The Hasse diagram of a poset P is the directed graph whose vertex set is the set ofelements of P such that for x, y ∈ P there is a directed edge from x to y if x is covered by y in P . A poset P is said to be connected if its Hasse diagram is connected. Clearly, if a finiteposet possesses a top element (denoted by ˆ1) or a bottom element (denoted by ˆ0), then it isconnected. A lattice is a poset L such that every pair of elements has a least upper bound3nd a greatest lower bound.The polar (or dual ) of a polytope P ⊂ Q m is the polytope P ◦ defined by P ◦ := { y ∈ ( Q m ) ∗ : h x, y i ≤ x ∈ P } . Here, h , i is the canonical evaluation pairing between Q m and ( Q m ) ∗ .Let x be a point in Q m , and let H be a hyperplane in Q m such that x / ∈ H . Let P bea polytope in H . The pyramid over P with apex at x is the convex hull conv( P , x ). Wewill denote a pyramid over P by pyr( P ).Let Q and P be two polytopes in Q m and Q n , respectively. The direct product (or simplythe product ) of Q and P , denoted by Q × P , is defined as the convex hull, Q × P := conv(( a, b ) : a ∈ V ( Q ) , b ∈ V ( P )) . We now assume that the origin of Q m (resp. of Q n ) is contained in Q (resp. in P ). The freesum of Q and P , denoted by Q ⊕ P , is defined as follows: Q ⊕ P := conv( Q × { Q n } , { Q m } × P ) . The purpose of this subsection is to introduce toric codes by circumventing much of the origi-nal definition of the algebraic-geometric codes. For a detailed introduction to this importantsubject, we recommend the textbook [15].Let N be a free abelian group of rank m , and let M denote its dual group. Let P be afull dimensional lattice polytope in M ⊗ Z Q . The lattice points in P ∩ M define monomialsthat are regarded as polynomial functions on the m -dimensional torus T N := Hom( N, F ∗ q ).Let H ( T N ( F q ) , P ) denote the F q -vector space that is spanned by these monomials. The toric code of P is then the image of the evaluation mapev : H ( T N ( F q ) , P ) −→ ( F ∗ q ) m f ( f ( x )) x ∈ T N ( F q ) . More generally, the algebraic-geometric code associated with an ample line bundle on anormal variety X that is defined over F q is the image of the germ-evaluation map on a setof F q -rational points S ⊆ X ( F q ). The toric codes from lattice polytopes are defined byevaluating on the F q -rational points of the open orbit of a normal toric variety.Hereafter, we denote by C P the toric code associated with a lattice polytope P . The length of C P is defined as length := ( q − m , where m is the dimension of the toric variety. The dimension of C P is defined as the vectorspace dimension of the space of sectionsdimension := dim H ( T N ( F q ) , P ) . P ∩ M . Finally, the computation of theminimum distance for the toric codes associated with an order polytope is the main focusof the present article. It is calculated as follows. For a section f ∈ H ( T N ( F q ) , P ), let Z ( f )denote the number of points in ( F ∗ q ) m where f vanishes. Then the minimum distance of C P ,denoted by d ( C P ), is given by d ( C P ) = ( q − m − max f ∈ H ( T N ( F q ) , P ) \{ } Z ( f ) . We will make use of the following results which are due to Soprunov and Soprunova.
Lemma 2.1. (Theorem 2.1 [13]) Let P and Q be two lattice polytopes contained in the boxes [0 , q − m ⊆ Q m and [0 , q − n ⊆ Q n , respectively. Then the minimum distance of the codeof the product P × Q is given by d ( C P × Q ) = d ( C P ) d ( C Q ) . Lemma 2.2. (Theorem 2.3 [13]) Let Q be a lattice polytope of dim Q ≥ . If P denotes theunit pyramid over Q , then we have d ( C P ) = ( q − d ( C Q ) . Let P = { ε , . . . , ε m } be a finite poset, and let N denote the free Z -module generated by P .Let ˆ P denote P ∪ { ε , ε m +1 } , where ε (resp. ε m +1 ) is such that ε (cid:12) ε i (resp. ε i (cid:12) ε m +1 )for every i ∈ { , . . . , m } . Let M denote the dual of N , that is M := Hom Z ( N, Z ), and let { e , . . . , e m } be the basis of M that is dual to P . For each covering relation ε i ⋖ ε j in ˆ P , weintroduce a vector ρ ( ε i , ε j ) in M ⊗ Z Q as follows: ρ ( ε i , ε j ) := e i if ε j = ˆ1; e i − e j if ε i , ε j ∈ P ; − e j if ε i = ˆ0. (3.1)The poset polytope of P , denoted by H P , is the convex hull of points ρ ( ε i , ε j ), where ε i ⋖ ε j is a cover in ˆ P . A systematic study of these polytopes is initiated by Hibi and Higashitaniin [7]. In this article, we construct linear error correcting codes by using (the polars of the)poset polytopes.Next, we will discuss poset polytopes and their relationship to the order polytopes. Sinceit is already introduced (in the Introduction), we will not repeat the definition of a posetpolytope here. In [7], Hibi and Higashitani showed that these polytopes have some remark-able properties. We will summarize the relevant results from [7] in the form of a single lemmato ease our referencing. Lemma 3.2.
For every poset P , the following statements hold:1. H P is a Fano polytope , that is, 0 is the unique integral interior point.2. H P is terminal , that is, each integral point on the boundary of H P is a vertex. . H P is Gorenstein , that is, its dual polytope is integral.4. If P is a graded poset of length l − , then the polar polytope of H P is the dilated andtranslated order polytope l O P − v , where v is the unique lattice point in l O P . The first item is proved in [7, Lemma 1.3], the second item is proved in [7, Lemma 1.4].The third item is proved in [7, Lemma 1.5]. The last item is recorded in [7, Remark 1.6]; itsproof follows from the definitions.
Remark 3.3.
A Gorenstein and Fano polytope is known as the reflexive polytope . In par-ticular, the dual of a reflexive polytope is reflexive. The normal fan of a reflexive polytopegives a “Gorenstein Fano toric variety” [1, Theorem 8.3.4]. (Such toric varieties are alwaysnormal.) In particular, a reflexive polytope is very ample in the sense of [1, Definition 2.2.17].
Notation 3.4. If P is a graded poset of length l −
2, then the polytope l O P − v , where v is the unique lattice point in l O P , will be denoted by O P ( l ). Example 3.5.
Let P (resp. ˆ P ) be the poset whose Hasse diagram is on the left (resp. onthe right) in Figure 3.1. Pε ε ε ˆ P ˆ0 ε ε ε ˆ1 Figure 3.1By fixing { ε , ε , ε } as a basis for N ⊗ Z Q , we will identify the elements of N ⊗ Z Q bytheir coordinate vectors. Then, the vertex set of O P consists of the following vectors in Q : ρ ( ∅ ) = (0 , , ,ρ ( { ε } ) = ε = (1 , , ,ρ ( { ε , ε } ) = ε + ε = (1 , , ,ρ ( { ε , ε } ) = ε + ε = (1 , , ,ρ ( { ε , ε , ε } ) = ε + ε + ε = (1 , , . In Figure 3.2, we depicted the order polytope of P . Finally, let us consider the dual polytopefor O P (3). It is easy to check that the vertices of the dual polytope H P are given by − e , e − e , e − e , e , e . We notice that the convex hull of e − e , e − e , e , e is arectangular plate, which we denote by A . Then H P is a pyramid over A with apex at − e .6 ••• • ε ε ε (0 , , , , , , , ,
1) (1 , , Figure 3.2: The order polytope of P .We close this subsection by two simple observations. Lemma 3.6.
Let P be a poset with connected components P , . . . , P r . Then we have H P = H P ⊕ · · · ⊕ H P r . Proof.
Let x be a vertex in H P . Then there is a covering relation ε i ⋖ ε j in ˆ P such that x ∈ { e i , e i − e j , − e j } . Since every covering relation in ˆ P is a covering relation in one of the posets ˆ P i ( i ∈ { , . . . , r } ),we see that the vertex set of H P is a disjoint union, V ( H P ) = V ( H P ) ⊔ · · · ⊔ V ( H P r ) . Note that, the subpolytopes H P i for i ∈ { , . . . , r } are contained in skew subspaces in Q m .Nevertheless, they all share the origin of Q m . Therefore, we have H P = conv( V ( H P ))= conv( V ( H P ) ⊔ · · · ⊔ V ( H P r ))= conv( V ( H P )) ⊔ · · · ⊔ conv( V ( H P r )) . This finishes the proof of our assertion.Our next observation is about the order polytopes.7 emma 3.7.
Let P be a poset with connected components P , . . . , P r . Then we have O P = O P × · · · × O P r . Proof.
Let x be a vertex in O P ⊆ Q m , where m is the number of elements of P . Then thereis an upper order ideal I in P such that x = ρ ( I ). Since P is the disjoint union P ⊔ · · · ⊔ P r ,we see that I = I ⊔ · · · ⊔ I r , where I i ( i ∈ { , . . . , r } ) is an upper order ideal in P i . It followsthat x is of the form x = x + · · · + x r ∈ Q m ⊕ · · · ⊕ Q m r , (3.8)where x i = ρ ( I i ), and Q m i is the vector subspace of Q m that is spanned by the basis vectorscorresponding to the elements of P i ( i ∈ { , . . . , r } ). The decomposition in (3.8) shows thatthe vertex set of O P is the product of the vertex sets of the order polytopes O P i , V ( O P ) = V ( O P ) × · · · × V ( O P r ) . This finishes the proof.The decompositions that we observed in Lemmas 3.7 and 3.6 can be obtained from eachother by induction and the well-known polarity correspondence between the free sums anddirect products of polytopes.
Remark 3.9.
As we mentioned in the introduction, a desirable code is the one with ahigh transmission rate , that is, dimension / length. The construction of H P uses the coverrelations in P whereas the construction of O P uses all upper order ideals in P . In generalthe vertices of the latter polytope is much more numerous. Therefore, for a generic poset P ,the transmission rate of C H P is very small compared to the transmission rate of C O P . We begin with a reduction result.
Proposition 4.1.
Let P be a poset with r connected components P , . . . , P r . Let q be aprime power such that q > . Then the minimum distance of the toric code C O P is given by d ( C O P ) = d ( C O P ) · . . . · d ( C O Pr ) . Proof.
We know from Lemma 3.7 that O P decomposes as a direct product, O P = O P × · · · × O P r . By applying induction with Lemma 2.1, we see that d ( C O P ) = d ( C O P ) · . . . · d ( C O Pr ).Next, we focus on the connected posets. 8 roposition 4.2. Let P = { ε , . . . , ε m } be a connected poset with a unique minimal element, ε . If P ′ is the poset obtained from P by removing ε , then we have d ( C O P ) = ( q − d ( C O P ′ ) . Proof.
Since ε is the smallest element in P , the upper order ideal generated by ε is thewhole poset P . In particular, all coordinates of the corresponding vertex x := ρ ( P ) in Q m is 1, x = (1 , . . . , ∈ Q m . For every other vertex x = ( a , . . . , a m ) of O P such that x = x , we have a = 0. This meansthat the line segment between vertices x and x is an edge of the polytope O P . (Note thatthis observation follows from [8, Lemma 1.1 (a)] as well.) It follows that O P is a pyramidover O P ′ . Now, the rest of the proof follows from Lemma 2.2.Let P be a poset. We call P a rooted tree poset if the following conditions hold:1. the Hasse diagram of P is a rooted tree, where the smallest element of P is the root;2. the leaves of P are the maximal elements of P .If P is the graded tree poset whose Hasse diagram is as in Figure 4.1, then we call it the m -th shrub . The m -th shrub will be denoted by S m . If the number m is understood fromthe context, or if it is not relevant to the discussion, then we simply write “shrub” insteadof writing “the m -th shrub.” Let I be an upper order ideal in S m . If I contains the element ε ε ε · · · ε m − ε m Figure 4.1: The m -th shrub, S m . ε , then it is equal to S m . If ε / ∈ I , then I can be any subset of { ε , . . . , ε m } . Therefore, J ( S oppm ) is isomorphic to B m − ⊕ ˆ1, where B m − is the boolean algebra of rank m −
1. Theproof of the following lemma is easy so we omit it.
Lemma 4.3.
Let m ≥ . Then the order polytope of the shrub S m is a pyramid over theunit cube of dimension m − . Next, we introduce the notion of a shrubbery of a tree poset P . Clearly, every leaf in P belongs to a unique shrub in P . For example, consider the tree poset in Figure 4.2. The treeposet in that figure has 4 subshrubs, whose Hasse diagrams are drawn in solid black lines.The shrubbery of P is the collection of subshrubs of P that contain the leaves of P .9 ••• • • • ••• •• • •• • • • ••• Figure 4.2: The shrubbery of a tree.
Theorem 4.4.
Let P = { ε , . . . , ε m } be a tree poset whose shrubbery consists of the shrubs, S m , . . . , S m s . Then the minimum distance of the code C O P is given by d ( C O P ) = ( q − m − P si =1 ( m i − ( q − P si =1 ( m i − . Proof.
By Proposition 4.2, the minimum distance C O P is equal to ( q − d ( C O P ′ ), where P ′ is agraded forest with connected components P , . . . , P r . By repeatedly applying Proposition 4.1and Proposition 4.2, we reach to the shrubberies of P i ’s, d ( C O P ) = ( q − j d ( C O Sm ) · . . . · d ( C O Smt ) , (4.5)where P ti =1 ( m i −
1) = s , hence, j + t = m − s . In this final step, we observe that, for each i ∈ { , . . . , t } , the order polytope O S mi is a pyramid over a unit cube of dimension m i − q − q − m i − . Thus, by substituting these into (4.5) we obtain the asserted formulafor the minimum distance C O P . Let P and Q be two posets. The ordinal sum of P and Q , denoted by P ⊕ Q , is the posetdefined on the disjoint union P ⊔ Q as follows. Let a and b be two elements from P ⊔ Q .Then a ≤ b ⇐⇒ if both of a and b are the elements of P , and a ≤ b in P ;if both of a and b are the elements of Q , and a ≤ b in Q ;if a ∈ P and b ∈ Q. The order polytope of the ordinal sum of two posets can be described in terms of theorder polytope of the summands.
Lemma 5.1.
Let P and Q be two posets. Then the order polytope of the ordinal sum P ⊕ Q is monomially equivalent to the free sum of polytopes O P ⊕ O Q .Proof. Let n and m denote the cardinalities of P and Q respectively. Then O P ⊂ Q n and O Q ⊂ Q m . Let I (resp. I ′ ) be an element of J ( P opp ) (resp. of J ( Q opp )). By abuse of10otation, we will use the same notation I (resp. I ′ ) for the upper order ideal generated by I (resp. I ′ ) in P ⊕ Q . In this notation, clearly, for every upper order ideal I of P we have Q ≤ I in J (( P ⊕ Q ) opp ). In terms of cartesian coordinates on Q n × Q m , this fact amountsto the fact that ρ P ⊕ Q ( I ) has 1’s on its last m coordinates. In other words, in Q n × Q m , thevector v := (0 , . . . , , , . . . ,
1) corresponds to both of 1) the empty upper order ideal of P ,2) the maximal upper order ideal of Q . We now consider the affine translate O P ⊕ Q − v in Q n × Q m . Under this translation, the vertices that correspond to the upper order ideal in P are mapped to the negatives of the lower order ideals in P . Therefore, we have the followingequality of polytopes: O P ⊕ Q − v = ( − O P opp ) ⊕ O Q . But the polytope − O P opp is monomially equivalent to O P , hence, we obtain the equivalence, O P ⊕ Q − v ∼ = O P ⊕ O Q . This finishes the proof of our assertion.Recall that the minimum distance of the toric code that is obtained from the directproduct of two polytopes P (in Q m ) and Q (in Q n ) is given by the product of the minimumdistances of the codes that are associated with P and Q (Lemma 2.1). Let h be a polynomialfrom H ( T N ( F q ) , P ). The weight of h , denoted wt ( h ), is the maximum number of nonzerocoordinates in the image vectors of the evaluation of h on the points of T N ( F q ). Let f be apolynomial from H ( T N ( F q ) , P ) such that wt ( f ) = d ( C P ). Similarly, let g be a polynomialfrom H ( T N ′ ( F q ) , Q ) such that wt ( g ) = d ( C Q ). In their proof of Lemma 2.1, Soprunov andSoprunova [13, Theorem 2.1] show that the weight of the polynomial f g is equal to d ( C P × Q ).Note that f and g separately belong also to the space of sections H ( T N × N ′ ( F q ) , P ⊕ Q ).This in particular gives us an upper bound for d ( C P ⊕ Q ) as follows. Clearly, the total numberof points in T N × N ′ ( F q ) ( ∼ = ( F ∗ q ) m + n ) where f (resp. g ) vanishes is given by Z ( f )( q − n (resp. by Z ( g )( q − m ). Thus, we have d ( C P ⊕ Q ) ≤ max { ( q − m + n − Z ( f )( q − n , ( q − m + n − Z ( g )( q − m } . Next, we apply this observation to an ordinal sum of posets.Let m be a positive integer. Let us denote an antichain with m elements by A m . Theorder polytope of A m is the m -dimensional unit cube. Note that an m -chain is given by A ⊕ · · · ⊕ A ( m copies), which we denote by C m . Lemma 5.2.
Let m be a positive integer. Then the minimum distance of the toric codeassociated with O A m ⊕ A m is given by ( q − m ( q − m .Proof. We begin with a slightly more general setup. Let m ≤ n be two positive integers. Weconsider the ordinal sum A m ⊕ A n . By Lemma 5.1, we have O A m ⊕ A n = O A m ⊕ O A n . Let f be a polynomial in H ( T N ( F q ) , O A m ) such that wt ( f ) = d ( C O Am ). Then we know that Z ( f ) = ( q − m − d ( C O Am ) = ( q − m − ( q − m . g be a polynomial in H ( T N ′ ( F q ) , O A n ) such that wt ( g ) = d ( C O An ). Then weknow that Z ( g ) = ( q − n − d ( C O An ) = ( q − n − ( q − n . Therefore, the minimum distance of O A m ⊕ A n is bounded by d ( C O Am ⊕ O An ) ≤ max { ( q − m + n − (( q − m − ( q − m )( q − n , ( q − m + n − (( q − n − ( q − n )( q − m } = max { ( q − m ( q − n , ( q − n ( q − m } = ( q − m ( q − n . In particular, if m = n , then we see that d ( C O Am ⊕ O An ) ≤ ( q − m ( q − m . (5.3)We notice that the poset A m ⊕ A m is covered by m H m := ⊔ mi =1 C . It is easy tocheck the polytope containment O A m ⊕ A m ⊆ O H m . Therefore, by our first main result, the minimum distance of our code C O Am ⊕ O Am is boundedfrom below by the minimum distance of H m , which is equal to ( q − m ( q − m . The rest ofthe proof follows from (5.3). Theorem 5.4.
Let m be a positive integer. The minimum distance of a toric code associatedwith an ( m, m ) -bipartite poset is given by ( q − m ( q − m .Proof. Let H m denote ⊔ mi =1 C . By the proof of Lemma 5.2, we know that d ( C O Am ⊕ Am ) = d ( C O Hm ) = ( q − m ( q − m . It is easy to check (by computing the vertices of the order polytopes) that if P is an ( m, m )-bipartite poset, then O A m ⊕ A m ⊆ O P ⊆ O H m . These inclusions give the following inequalities: d ( C O Am ⊕ Am ) ≥ d ( C O P ) ≥ d ( C O Hm ) , which are actually equalities. This finishes the proof of our theorem. Proposition 5.5.
Let m be a positive integer. Then we have the following formulas for thedimensions of the toric codes associated with A m ⊕ A m and H m := ⊔ mi =1 C .1. dim C O Am ⊕ Am = 2 m +1 − , and2. dim C O Hm = 3 m . ( P opp ) J ( P opp ) J ( P opp ) { ε , ε , ε , ε } { ε , ε , ε , ε } { ε , ε , ε , ε }{ ε , ε , ε } { ε , ε , ε } { ε , ε , ε }{ ε , ε , ε } { ε , ε , ε } { ε , ε , ε }{ ε , ε } { ε , ε } { ε , ε }{ ε } { ε , ε } { ε , ε }{ ε } { ε } { ε , ε }∅ { ε } { ε }∅ { ε }∅ Table 1: The upper order ideals of P , P , P . Proof.
The dimension of a toric code defined by an order polytope is equal to the numberof vertices of the polytope. In the former case, we have the free sum of two m dimensionalcubes. Therefore, the dimension in this case is given by 2 m + 2 m − m +1 −
1. In thelatter case, the vertices of O H m are given by the upper order ideals in H m . Any such idea isuniquely determined by a minimum elements ˆ0 i , . . . , ˆ0 i a in H m , and b maximum elementsˆ1 j , . . . , ˆ1 j b , where ˆ1 j r (1 ≤ r ≤ b ) does not cover any element from { ˆ0 i , . . . , ˆ0 i a } . Therefore,the total number of such upper order ideals is given by P ma =0 P m − ab =0 (cid:0) ma (cid:1)(cid:0) m − ab (cid:1) . By using thebinomial theorem, we see that this sum is equal to 3 m . Example 5.6.
We consider the posets P , P and P that are defined in Figure 5.1. InTable 1 we listed their upper order ideals. ε ε ε ε ε ε ε ε ε ε ε ε Figure 5.1: The posets P , P , and P (from left to right).The minimum distance of the toric code associated with the order polytope of P i ( i ∈{ , , } ) equals d ( C O Pi ) = ( q − ( q − . Acknowledgement
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