An Algebraic-Combinatorial Proof Technique for the GM-MDS Conjecture
aa r X i v : . [ c s . I T ] M a y An Algebraic-Combinatorial Proof Technique forthe GM-MDS Conjecture
Anoosheh Heidarzadeh and Alex SprintsonTexas A&M University, College Station, TX 77843 USA
Abstract —This paper considers the problem of designingmaximum distance separable (MDS) codes over small fields withconstraints on the support of their generator matrices. For anygiven m × n binary matrix M , the GM-MDS conjecture , due toDau et al. , states that if M satisfies the so-called MDS condition,then for any field F of size q ≥ n + m − , there exists an [ n, m ] q MDS code whose generator matrix G , with entries in F , fits M (i.e., M is the support matrix of G ). Despite all the attempts bythe coding theory community, this conjecture remains still open ingeneral. It was shown, independently by Yan et al. and Dau et al. ,that the GM-MDS conjecture holds if the following conjecture,referred to as the TM-MDS conjecture , holds: if M satisfies theMDS condition, then the determinant of a transformation matrix T , such that T V fits M , is not identically zero, where V is aVandermonde matrix with distinct parameters. In this work, wegeneralize the TM-MDS conjecture, and present an algebraic-combinatorial approach based on polynomial-degree reductionfor proving this conjecture. Our proof technique’s strength isbased primarily on reducing inherent combinatorics in the proof.We demonstrate the strength of our technique by proving theTM-MDS conjecture for the cases where the number of rows( m ) of M is upper bounded by . For this class of special casesof M where the only additional constraint is on m , only cases with m ≤ were previously proven theoretically, and the previouslyused proof techniques are not applicable to cases with m > . I. I
NTRODUCTION
In recent years, there has been a growing interest indesigning maximum distance separable (MDS) codes withconstraints on the support of the codes’ generator matrix[1]–[11]. Such constraints arise in wireless network codingand distributed storage scenarios where each user/server hasaccess to only a subset of the information symbols. Twoexamples of such scenarios are cooperative data exchange inthe presence of an eavesdropper [1], [5], and simple multipleaccess networks with link/relay errors [6], [8].Given an m × n binary (support) matrix M = ( M i,j ) anda field F of size q , the problem is to design an [ n, m ] q MDScode with a generator matrix G = ( G i,j ) , G i,j ∈ F (i.e.,all m × m sub-matrices of G are full-rank) fitting M (i.e.,if M i,j = 0 , then G i,j = 0 ). Note that for some M , thereexists no MDS code whose generator matrix fits M (i.e., M is not completable to MDS). Nevertheless, there is a simplecondition, known as the MDS condition, which characterizesall matrices that are completable to MDS for sufficiently largefields [4], [5]. A matrix M satisfies the MDS condition if: |∪ i ∈ I supp( M i ) |≥ n − m + | I | , ∀ I ⊆ { , . . . , m } , I = ∅ , where M i is the i th row of M , and supp( M i ) is the supportof M i . The existence of MDS codes (over sufficiently large fields) whose generator matrix’s support satisfies the MDScondition was shown, e.g., in [8], via Edmonds matrix andHall’s marriage theorem. The following conjecture, due toDau et al. [4], aims for generalizing this result for small fields. Conjecture 1 (GM-MDS Conjecture):
If the matrix M sat-isfies the MDS condition, then for any field F of size q ≥ n + m − , there exists an [ n, m ] q MDS code whose generatormatrix G with entries in F fits the matrix M .Notwithstanding all the efforts by the coding theory com-munity, the GM-MDS conjecture remains still open in general.The GM-MDS conjecture and a simplified version of thisconjecture where the supports of rows of M have the samesize were shown in [8] to be equivalent (using a generalizedversion of Hall’s theorem). Despite this simplification, thereare only three classes of special cases for which this conjectureis theoretically proven: (i) the rows of M are divided into threegroups, and the rows in each group have the same support [6];(ii) the size of intersection of the supports of every two rowsof M is upper bounded by [8]; and (iii) the number ofrows of M is upper bounded by [11]. More importantly, thepreviously used proof techniques are not applicable to moregeneral cases due to the combinatorial explosion.One possible approach to find a completion G of M toMDS is to leverage the structure of Generalized Reed-Solomon(GRS) codes [4], [5] which are known to be MDS. Let N bethe set of n independent indeterminates α , . . . , α n . Let M bean m × n binary matrix whose rows’ supports have the samesize, and let V = V ( N ) be a generic m × n Vandermondematrix with parameters α , . . . , α n . Let T = T ( M, N ) be ageneric m × m transformation matrix such that T V fits M ,and let G = T V . If the evaluations α ∗ , . . . , α ∗ n of α , . . . , α n are distinct, then every m × m sub-matrix of G is non-singular(i.e., G is a generator matrix of a GRS code with evaluationpoints α ∗ , . . . , α ∗ n ) so long as T is non-singular. That is,if T is not generically singular (i.e., the determinant of T as a multivariate polynomial in variables α , . . . , α n is notidentically zero), then for any field F of size q ≥ n + m − ,there exists α ∗ , . . . , α ∗ n ∈ F such that T V is a generator matrixof an [ n, m ] q MDS code, and
T V fits M . Thus, the GM-MDS conjecture holds if the following conjecture, proposedindependently by Dau et al. [4] and Yan et al. [5], holds: Conjecture 2 (TM-MDS Conjecture): If M satisfies theMDS condition, then T ( M, N ) is not generically singular.The contributions of this work are as follows. First, wepresent a generalization of the TM-MDS conjecture for thecases where the supports of rows of M have arbitrary sizes.hen, we present an algebraic-combinatorial approach basedon polynomial-degree reduction for proving this conjecture.Our technique’s strength is primarily due to reducing the in-herent combinatorics in the proof. Specifically, we demonstratethis strength by proving the TM-MDS conjecture for the caseswhere the number of rows of M is upper bounded by .II. B ASIC N OTATIONS AND D EFINITIONS
Let F be a field. For n ∈ N , let α , . . . , α n be n independentindeterminates. Let F [ α , . . . , α n ] be a ring of multivariatepolynomials in variables α , . . . , α n with coefficients in F , andlet ( F [ α , . . . , α n ])[ α ] be a module of univariate polynomialsin variable α with coefficients in F [ α , . . . , α n ] . Fix m ∈ N and n ∈ N such that < m ≤ n ≤ m ( m − . For k ∈ N ,denote { , . . . , k } by [ k ] . Define m polynomials P , . . . , P m of degrees ≤ d , . . . , d m ≤ m − in ( F [ α , . . . , α n ])[ α ] : P i ( α ) , Y γ ∈ N i ( α − γ ) , ∀ i ∈ [ m ] , (1)where N i , the set of roots of P i , is a (proper) subset of N , { α , . . . , α n } of size d i . (Note that the roots of P i areindeterminates.) For N i = ∅ and d i = 0 , P i ( α ) , .Note that P i ( α ) = P j ∈ [ m ] C i,j α j − , where { C i,j } j ∈ [ m ] are polynomials in F [ α , . . . , α n ] . Define W ( P , . . . , P m ) , det(( C j,i ) i,j ∈ [ m ] ) . Note that W ( P , . . . , P m ) is a polynomialin F [ α , . . . , α n ] . Definition 1:
A polynomial W in F [ α , . . . , α n ] is iden-tically zero , denoted by W ≡ , if the coefficients of allmonomials in the polynomial expansion of W are zero. Definition 2:
A set { P , . . . , P m } of m polynomials of de-gree m − has rectangular property (RP) if, for some < k ≤ m , there exist at least k polynomials in { P , . . . , P m } with atleast m − k + 1 common roots. Otherwise, { P , . . . , P m } has non-rectangular property (NRP). Definition 3:
A set { P , . . . , P m } of m polynomials ofdegrees ≤ d , . . . , d m ≤ m − has generalized RP (GRP) if,for some < k ≤ m and ≤ l ≤ m − k , there exist at least k polynomials of degrees at most m − l − in { P , . . . , P m } withat least m − k − l + 1 common roots. Otherwise, { P , . . . , P m } has generalized NRP (GNRP).III. M AIN C ONJECTURES AND T HEOREMS
The following conjecture is equivalent to the TM-MDSconjecture (Conjecture 2).
Conjecture 3:
Let P , . . . , P m be m polynomials of degree m − in ( F [ α , . . . , α n ])[ α ] . If W ( P , . . . , P m ) ≡ , then { P , . . . , P m } has RP.The sketch of proof of the equivalency between the TM-MDS conjecture and Conjecture 3 follows. Let M = ( M i,j ) be an m × n binary matrix. Let M i be the i th row of M and let supp( M i ) be the support of M i . Let N i = { α j : j ∈ [ n ] \ supp( M i ) } , where α , . . . , α n are n independentindeterminates. Suppose that all N i have the same size.Defining P i ( = P i ( α ) ) as in (1), it follows that the matrix M satisfies the MDS condition iff { P , . . . , P m } has NRP(refer to this as Fact 1). Let G = ( G i,j ) be a generic m × n generator matrix of a Generalized Reed-Solomon (GRS) codewith evaluation points α , . . . , α n such that G fits M (i.e.,if M i,j = 0 , then G i,j = 0 ). Let V = ( V i,j ) be a generic m × n Vandermonde matrix with parameters α , . . . , α n , andlet T = ( T i,j ) be a generic m × m transformation matrixsuch that G = T V . Taking V i,j = α j − i and T i,j = C j,i ,where P i ( α ) = P j ∈ [ m ] C i,j α j − , it follows that G = T V (for more details, see [5]). Since det( T ) = W ( P , . . . , P m ) ,then det( T ) (i.e., T is not generically singular) iff W ( P , . . . , P m ) (refer to this as Fact 2). By Facts 1 and2, the TM-MDS conjecture and Conjecture 3 are equivalent.In the following, we propose a new conjecture which gener-alizes Conjecture 3 for the cases where the degrees d , . . . , d m of polynomials P , . . . , P m are arbitrary. (Conjecture 4 isequivalent to a generalized version of the TM-MDS conjecturewhere the supports of rows of M have arbitrary sizes.) Conjecture 4:
Let P , . . . , P m be m polynomials of arbi-trary degrees ≤ d , . . . , d m ≤ m − in ( F [ α , . . . , α n ])[ α ] .If W ( P , . . . , P m ) ≡ , then { P , . . . , P m } has GRP.If d i < m − for all i ∈ [ m ] , then Conjecture 4 holdstrivially: (i) W ( P , . . . , P m ) ≡ since C i,m = 0 for all i ∈ [ m ] , and (ii) { P , . . . , P m } has GRP since for k = m and l = 1 , there exist k polynomials of degrees at most m − l − in { P , . . . , P m } with at least m − k − l + 1 common roots.Hereafter, w.l.o.g., we assume d i = m − for some i ∈ [ m ] .The following theorems, which are our main results, provethe GM-MDS conjecture for m ≤ . More specifically,Theorems 1, 2, and 3 settle Conjecture 4 (and so Conjecture 3)for m ≤ , and Theorem 4 settles Conjecture 3 for m = 5 . Theorem 1:
For any P , P such that ≤ d ≤ d = 1 , if W ( P , P ) ≡ , then { P , P } has GRP. Theorem 2:
For any P , P , P such that ≤ d ≤ d ≤ d = 2 , if W ( P , P , P ) ≡ , then { P , P , P } has GRP. Theorem 3:
For any P , . . . , P such that ≤ d ≤ . . . ≤ d = 3 , if W ( P , . . . , P ) ≡ , then { P , . . . , P } has GRP. Theorem 4:
For any P , . . . , P such that d = · · · = d =4 , if W ( P , . . . , P ) ≡ , then { P , . . . , P } has GRP.IV. M AIN I DEAS AND L EMMAS
In this section, we explain the main ideas and state theuseful lemmas for the proofs of our main results.Consider an arbitrary set { P i } ( = { P i } ≤ i ≤ m ) of m poly-nomials P i (with the sets of roots N i ) such that ≤ d i ≤ m − for all i ∈ [ m ] , and d i = m − for some i ∈ [ m ] . Define a classof reduction processes over { P i } , where any process in thisclass is associated with a unique reduction set R ⊆ N , and itreduces P i ( α ) = Q γ ∈ N i ( α − γ ) to ˜ P i ( α ) , Q γ ∈ N i \ R ( α − γ ) .Let ˜ d i , deg( ˜ P i ) . Note that ˜ d i = d i − | N i ∩ R | . Restrict yourattention to those reduction sets R such that ˜ d j = m − forsome j ∈ [ m ] , and ˜ d i < m − for all i ∈ [ m ] \ { j } . Such R are referred to as acceptable . For any acceptable reductionset, w.l.o.g., assume that ˜ d i < m − for all i ∈ [ m − and ˜ d m = m − (and so, d m = m − since ˜ d m ≤ d m ≤ m − ).For any (acceptable) reduction set, the following result holds. Lemma 1: If W ( P , . . . , P m ) ≡ , then W ( ˜ P , . . . , ˜ P m − ) ≡ . roof: Consider the resulting { ˜ P i } from { P i } for anarbitrary (acceptable) reduction set R = { α r , . . . , α r | R | } .Let r R , { r , . . . , r | R | } . For any r ∈ r R , let n r be thenumber of polynomials P i in { P i } such that α r ∈ N i , andlet n R , { n r , . . . , n r | R | } . Let W ( n R ) ( P , . . . , P m ) be theresulting polynomial from W ( P , . . . , P m ) by taking deriva-tive n r times with respect to each variable α r ∈ R . (Since W ( P , . . . , P m ) = det(( C j,i ) i,j ∈ [ m ] ) , and C j,i is the sum ofmonomials ( − e + ··· + e n α e · · · α e n n for some { e , . . . , e n } ∈{ , } n (depending on i, j ), then the derivatives of C j,i withrespect to any variable α r are independent of F .) Notethat W ( n R ) ( P , . . . , P m ) = ( − n r + ··· + n r | R | W ( ˜ P , . . . , ˜ P m ) (by using the Leibniz formula for determinant), and W ( ˜ P , . . . , ˜ P m ) = W ( ˜ P , . . . , ˜ P m − ) (since ˜ d i < m − and ˜ C i,m = 0 for all i ∈ [ m − , and ˜ d m = m − and ˜ C m,m = 1 ,where ˜ P i ( α ) = P j ∈ [ m ] ˜ C i,j α j − ). Since W ( P , . . . , P m ) ≡ (by assumption), then W ( n R ) ( P , . . . , P m ) ≡ . Thus, W ( ˜ P , . . . , ˜ P m − ) ≡ .Lemma 1 enables us to use an inductive argument towardsthe proof of Conjecture 4 as follows. Suppose that Conjec-ture 4 holds for any < l ≤ m − , i.e., for any { P , . . . , P l } such that ≤ d i ≤ l − for all i ∈ [ l ] and d i = l − for some i ∈ [ l ] , if W ( P , . . . , P l ) ≡ , then { P , . . . , P l } has GRP. Weneed to prove that for any { P , . . . , P m } such that ≤ d i ≤ m − for all i ∈ [ m ] and d i = m − for some i ∈ [ m ] , if W ( P , . . . , P m ) ≡ , then { P , . . . , P m } has GRP. The prooffollows by contradiction. Assume that W ( P , . . . , P m ) ≡ and { P , . . . , P m } does not have GRP. Consider the resulting { ˜ P , . . . , ˜ P m } from { P , . . . , P m } for an (acceptable) reduc-tion set such that { ˜ P , . . . , ˜ P m − } has GNRP. By definition, ˜ d i < m − for all i ∈ [ m − . Since W ( P , . . . , P m ) ≡ (byassumption), then W ( ˜ P , . . . , ˜ P m − ) ≡ (by Lemma 1), andso, { ˜ P , . . . , ˜ P m − } has GRP (by the induction hypothesis),yielding a contradiction. Our goal is thus to devise an (accept-able) reduction process such that if { P , . . . , P m } has GNRP,then so does { ˜ P , . . . , ˜ P m − } . The problem of designing sucha process is still open in general. In the following, we proposea simple yet powerful reduction process which solves thisproblem for m ≤ and ≤ d i ≤ m − for all i ∈ [ m ] ,and for m = 5 and d i = m − for all i ∈ [ m ] .From now on, we assume that { P i } ( = { P i } ≤ i ≤ m ) is aset of m polynomials P i (with the sets of roots N i ) such that ≤ d i ≤ m − for all i ∈ [ m − , and d m = m − . Definition 4:
A subset S ⊂ N is an ( r, s ) -subset in a subset Q of { P i } if S belongs to r polynomials in Q (i.e., thereexist r polynomials P i in Q such that S ⊂ N i ), and | S | = s .Moreover, an ( r, s ) -subset has higher order than an ( r ⋆ , s ⋆ ) -subset if r + s > r ⋆ + s ⋆ , or r + s = r ⋆ + s ⋆ and r > r ⋆ .The following lemma gives the intuition behind the defini-tion of ( r, s ) -subsets. Lemma 2: If { P i } has GNRP, then there exists no ( r, s ) -subset in { P i } such that r + s > m . Proof:
The proof is straightforward and follows from thedefinitions (and hence omitted).Intuitively, for any (acceptable) reduction set R , any highest-order ( r, s ) -subset S , if not broken (i.e., S ∩ R = ∅ ), is the most likely to cause rectangularity in { ˜ P i } for any { P i } withnon-rectangular property. This is the main idea of the proposedreduction process. Definition 5:
An element β of a subset S ⊂ N is removable if β is a root of some but not all polynomials of degree m − . Definition 6:
A subset S ⊂ N is weakly reducible if S belongs to a polynomial of degree m − , and S has aremovable element. Definition 7:
A weakly reducible ( r, s ) -subset S is stronglyreducible if no other weakly reducible ( r ⋆ , s ⋆ ) -subset hashigher order than S . Proposed Reduction Process:
Given { P i } , choose an ar-bitrary strongly reducible subset S in { P i } , and choose anarbitrary removable element of S , say β , such that no otherremovable element of S , when compared to β , belongs to morepolynomials of degree m − in { P i } . Break S via removing β from the sets N i of roots of all polynomials P i , and updateall polynomials P i via replacing N i by N i \ { β } . Repeat thisprocess (in rounds) over the resulting { P i } if there exist morethan one polynomial of degree m − . Otherwise, terminatethe process, and return the resulting { P i } denoted by { ˜ P i } .Note that if { P i } has GNRP initially, then (i) in each roundof the process, such β exists, and (ii) the process terminateseventually. Otherwise, there must exist two (or more) identicalpolynomials of degree m − in { P i } (and hence { P i } hasGRP), which is a contradiction.Consider an arbitrary run of the reduction process over { P i } and its corresponding { ˜ P i } . Let R be the set of the roots thatthe reduction process removes over the rounds. (Note that,due to the arbitrary choices in the reduction process, R mayor may not be unique.) Hereafter, for any such R , assume,w.l.o.g., that the (initial) indexing of polynomials in { P i } issuch that ˜ d ≤ . . . ≤ ˜ d m − < ˜ d m ( = d m ) and ˜ P m = P m , anddenote { P i } ≤ i ≤ m − by { P i } ⋆ .The proofs of our main theorems rely on the followingproperties of the proposed reduction process. Lemma 3: If { P i } has GNRP, then any ( r, s ) -subset in { P i } ⋆ such that r + s = m belongs to a polynomial P i in { P i } ⋆ such that d i = m − . Proof:
Let S be an arbitrary ( r, s ) -subset in { P i } ⋆ suchthat r + s = m . Let Q be the set of all polynomials P i in { P i } ⋆ such that d i < m − . Note that Q is a set of at most m − polynomials of degrees at most m − . Since { P i } has GNRP (by assumption), then { P i } ⋆ (and hence Q ) hasGNRP. Thus, there exists no ( r ⋆ , s ⋆ ) -subset in Q such that r ⋆ + s ⋆ > m − (by Lemma 2). Suppose that S belongs to nopolynomial P i in { P i } ⋆ \ Q . Then, S is an ( r, s ) -subset in Q such that r + s = m . This is, however, a contradiction. Thus, S belongs to a polynomial P i in { P i } ⋆ \ Q . Lemma 4: If { P i } has GNRP, then any ( r, s ) -subset in { P i } ⋆ such that r + s = m is weakly reducible. Proof:
Let S be an arbitrary ( r, s ) -subset in { P i } ⋆ suchthat r + s = m . Note that S belongs to a polynomial in { P i } ⋆ (and so { P i } ) of degree m − (by Lemma 3). Note, also, that P m has degree m − . Thus, if there exists β ∈ S such that β N m , then S is weakly reducible since β is removable (byefinition). Otherwise, if S ⊆ N m , then S is an ( r + 1 , s ) -subset in { P i } . Since r + 1 + s = m + 1 > m , then { P i } hasGRP (by Lemma 2), yielding a contradiction. Lemma 5: If { P i } has GNRP, then the strongly reducible ( r, s ) -subsets in { P i } ⋆ such that r + s = m belong to disjointsubsets of { P i } ⋆ . Proof:
Let S and S be two arbitrary strongly reducible ( r, s ) -subsets in { P i } ⋆ such that r + s = m . Let Q or Q bethe set of r polynomials P i in { P i } ⋆ such that S or S belongsto P i , respectively. Note that ≤ |Q ∩Q |≤ r . First, supposethat |Q ∩ Q | = r . Then, S = S ∪ S is an ( r, | S ∪ S | ) -subset in { P i } ⋆ . Since r + | S ∪ S | > r + s = m , then { P i } hasGRP (by Lemma 2), and hence a contradiction. Next, supposethat < |Q ∩ Q | < r . We consider two cases. First, supposethat | S ∩ S | < m − r + |Q ∩ Q | . Then, S = S ∪ S is a ( |Q ∩Q | , s −| S ∩ S | ) -subset in { P i } ⋆ . Let r ⋆ = |Q ∩Q | and s ⋆ = 2 s − | S ∩ S | . Since r ⋆ + s ⋆ = |Q ∩ Q | + m − r −| S ∩ S | > m , then { P i } has GRP (by Lemma 2), which is acontradiction. Next, suppose that | S ∩ S |≥ m − r + |Q ∩Q | .Then, S = S ∩ S is a (2 r − |Q ∩ Q | , | S ∩ S | ) -subset in { P i } ⋆ . Let r ⋆ = 2 r − |Q ∩ Q | and s ⋆ = | S ∩ S | . Note that r ⋆ + s ⋆ = 2 r − |Q ∩ Q | + | S ∩ S |≥ m . If r ⋆ + s ⋆ > m ,then { P i } has GRP (by Lemma 2), and hence a contradiction.If r ⋆ + s ⋆ = m , then S is weakly reducible (by Lemma 4),and S has higher order than S and S since r ⋆ + s ⋆ = r + s and r ⋆ > r . This is also a contradiction since S and S arestrongly reducible (by assumption). Thus, |Q ∩ Q | = 0 . Lemma 6:
For m = 2 , , and ≤ d i ≤ m − for all i ∈ [ m ] , and for m = 5 and d i = m − for all i ∈ [ m ] ,if { P i } has GNRP, then the reduction process breaks anystrongly reducible ( r, s ) -subset in { P i } ⋆ such that r + s = m . Proof:
Let S be an arbitrary strongly reducible ( r, s ) -subset in { P i } ⋆ such that r + s = m . Since { P i } has GNRP, S belongs to a polynomial P i in { P i } ⋆ of degree m − (by Lemma 3) and no other strongly reducible ( r, s ) -subsetin { P i } ⋆ belongs to P i (by Lemma 5). Moreover, for any m = 2 , , and any ≤ d ≤ . . . ≤ d m = m − , there existsno other ( r, s ) -subset in { P i } ⋆ such that r + s = m (otherwise, { P i } has GRP). Thus S must be broken to reduce P i .For m = 5 and d = · · · = d = 4 , S is either a (4 , -or (3 , - or (2 , -subset in { P i } ⋆ . First, suppose that S is a (4 , - or (3 , -subset in { P i } ⋆ . Since there exists no other (4 , - or (3 , -subset in { P i } ⋆ (otherwise, { P i } has GRP),then S must be broken to reduce P i . Next, suppose that S isa (2 , -subset in { P i } ⋆ . Let Q be the set of two polynomialsin { P i } ⋆ , say P and P , such that S belongs to both P and P . Let T be an arbitrary (if any) strongly reducible (2 , -subset in { P i } ⋆ \ Q . If T does not exist, then S must bebroken to reduce both P and P . If T exists, no elementof T is a common root of both P and P (otherwise, thereexists a strongly reducible (4 , -subset in { P i } ⋆ , which is acontradiction since S is a strongly reducible (2 , -subset).Since { P i } has GNRP, then there exists no other stronglyreducible (2 , -subset in { P i } ⋆ (by Lemma 5), and breaking T cannot reduce both P and P simultaneously. Thus, S mustbe broken to reduce P or P (or both). V. P ROOFS OF M AIN T HEOREMS
In this section, we prove our main theorems. For simplicity,we denote the degree-set of polynomials P , . . . , P m and ˜ P , . . . , ˜ P m by ( d , . . . , d m ) and ( ˜ d , . . . , ˜ d m ) , respectively. Proof of Theorem : Assume that W ( P , P ) ≡ . If ( d , d ) = (0 , , then W ( P , P ) = 1 , which is acontradiction. If ( d , d ) = (1 , , then W ( P , P ) = P − P .Thus, P = P , i.e., { P , P } has RP (and hence GRP). Proof of Theorem : The proof follows by contradic-tion. Assume that W ( P , P , P ) ≡ , and { P , P , P } hasGNRP. If ( d , d , d ) = (2 , , , then ( ˜ d , ˜ d , ˜ d ) = (1 , , (since the reduction process either reduces both P and P simultaneously, or it first reduces one, and then reduces theother one). Since W ( P , P , P ) ≡ (by assumption), then W ( ˜ P , ˜ P ) ≡ (by Lemma 1). Thus, { ˜ P , ˜ P } has GRP (byTheorem 1), i.e., there exists a (2 , -subset S in { ˜ P , ˜ P } .Thus, S is a strongly reducible ( r, s ) -subset in { P , P } suchthat r + s = m = 3 (by Lemmas 4 and 2), and it must havebeen broken (by Lemma 6), yielding a contradiction.If ( d , d , d ) = (1 , , , then ( ˜ d , ˜ d , ˜ d ) ∈{ (1 , , , (0 , , } (since the reduction process mustreduce P , and reducing P may or may not reduce P ).If ( ˜ d , ˜ d , ˜ d ) = (1 , , , then { ˜ P , ˜ P } has GRP, yieldinga contradiction as before. If ( ˜ d , ˜ d , ˜ d ) = (0 , , , then W ( ˜ P , ˜ P ) = 1 . Thus, W ( P , P , P ) (byLemma 1), which is again a contradiction.If ( d , d , d ) = (0 , , , then ( ˜ d , ˜ d , ˜ d ) = (0 , , (since the reduction process must reduce P , and reducing P does not reduce P ). Since W ( ˜ P , ˜ P ) = 1 , then W ( P , P , P ) (by Lemma 1), yielding a contradiction.If ( d , d , d ) = (1 , , , then ( ˜ d , ˜ d , ˜ d ) = (1 , , (sincethe reduction process does not reduce P and P ). Thus, { ˜ P , ˜ P } ( = { P , P } ) has GRP (by the same argument asbefore), which is a contradiction. If ( d , d , d ) = (0 , , ,then W ( P , P , P ) = 1 , yielding a contradiction. If ( d , d , d ) = (0 , , , then P = P = 1 . Thus, { P , P } has GRP, again a contradiction. Proof of Theorem : Due to the lack of space, we onlygive the proofs for the cases of ( d , . . . , d ) ∈ { (3 , , , , (2 , , , , (1 , , , , (2 , , , } . (The proofs of the restof the cases follow the exact same lines.) The proof isby way of contradiction. Assume that W ( P , . . . , P ) ≡ ,and { P , . . . , P } has GNRP. Since W ( ˜ P , ˜ P , ˜ P ) ≡ (byLemma 1), then { ˜ P , ˜ P , ˜ P } has GRP (by Theorem 2).First, consider ( d , . . . , d ) = (3 , , , . By theprocedure of the reduction process, ( ˜ d , . . . , ˜ d ) ∈{ (2 , , , , (1 , , , } . Since { ˜ P , ˜ P , ˜ P } has GRP,either there exists a (3 , -subset S , or if S does not exist,there exists a (2 , -subset S , in { ˜ P , ˜ P , ˜ P } . Since S (or S ) is a strongly reducible ( r, s ) -subset in { P , P , P } suchthat r + s = m = 4 (by Lemmas 4 and 2), S (or S ) musthave been broken (by Lemma 6), which is a contradiction.Second, consider ( d , . . . , d ) = (2 , , , . Then, ( ˜ d , . . . , ˜ d ) ∈ { (2 , , , , (1 , , , , (0 , , , } . For any ofthese cases, by the same arguments as for the previous case,we arrive at a contradiction.ext, consider ( d , . . . , d ) = (1 , , , . Then, ( ˜ d , . . . , ˜ d ) ∈ { (1 , , , , (0 , , , , (0 , , , } . Forthe cases of ( ˜ d , . . . , ˜ d ) ∈ { (1 , , , , (0 , , , } ,similar to the previous cases, we reach a contradiction.For the case of ( ˜ d , . . . , ˜ d ) = (0 , , , , it follows that W ( ˜ P , ˜ P , ˜ P ) = 1 , which is again a contradiction.Lastly, consider ( d , . . . , d ) = (2 , , , . Then, ( ˜ d , . . . , ˜ d ) ∈ { (2 , , , , (1 , , , , (1 , , , } . For thecases of ( ˜ d , . . . , ˜ d ) ∈ { (2 , , , , (1 , , , } , following theexact same lines as above yields a contradiction. Now, considerthe case of ( ˜ d , . . . , ˜ d ) = (1 , , , . Since ˜ d = d − , ˜ d = d − , and ˜ d = d − , then reducing P musthave reduced P and P simultaneously. Thus, there existsa (3 , -subset { β } in { P , P , P } . Since { ˜ P , ˜ P , ˜ P } hasGRP, there also exists a (2 , -subset { β } in { ˜ P , ˜ P } .Thus, { P , P } has GRP since { β , β } is a (2 , -subset in { P , P } , yielding a contradiction. Proof of Theorem : The proof follows by contradic-tion. Assume that W ( P , . . . , P ) ≡ , and { P , . . . , P } has GNRP. Since W ( ˜ P , . . . , ˜ P ) ≡ (by Lemma 1),then { ˜ P , . . . , ˜ P } has GRP (by Theorem 3). By the proce-dure of the reduction process, ( ˜ d , . . . , ˜ d ) ∈ { (3 , , , , , (2 , , , , , (1 , , , , , (2 , , , , } . Consider any ofthe cases of ( ˜ d , . . . , ˜ d ) ∈ { (3 , , , , , (2 , , , , , (1 , , , , } . Since { ˜ P , . . . , ˜ P } has GRP, there exists eithera (4 , -subset S , or a (3 , -subset S (if S does notexist), or a (2 , -subset S (if neither S nor S exists), in { ˜ P , . . . , ˜ P } . Since S (or S or S ) is a strongly reducible ( r, s ) -subset in { P , . . . , P } such that r + s = m = 5 (byLemmas 4 and 2), it must have been broken by the reductionprocess (by Lemma 6), which is a contradiction.Now, consider the case of ( ˜ d , . . . , ˜ d ) = (2 , , , , .Since { ˜ P , . . . , ˜ P } has GRP, there exists either a (4 , -subset S , or a (3 , -subset S , or a (2 , -subset S , in { ˜ P , . . . , ˜ P } , or if neither of S , S , and S exists, there existsa (2 , -subset S in { ˜ P , ˜ P } . If S (or S or S ) exists, thenit is a strongly reducible ( r, s ) -subset in { P , . . . , P } suchthat r + s = m = 5 (by Lemmas 4 and 2), and it musthave been broken (by Lemma 6), yielding a contradiction. Ifneither of S , S , and S exists, but S exists, then thereexists a (2 , -subset { β , β } in { ˜ P , ˜ P } . Since ˜ d = d − , ˜ d = d − , ˜ d = d − , and ˜ d = d − , either (i) P and P are reduced separately, and reducing P and reducing P both have reduced P and P simultaneously, or (ii) P and P are reduced simultaneously (without reducing P or P ), and reducing P has reduced P (or P ) but not P , andreducing P has reduced P (or P ) but not P .First, consider the case (i). Since reducing P has re-duced both P and P , there exists a (3 , -subset { β } in { P , P , P } such that β = β , β . Similarly, there exists a (3 , -subset { β } in { P , P , P } such that β = β , β . Notethat β = β since otherwise, { β } or { β } is a (4 , -subsetin { P , . . . , P } , which is a contradiction. Thus, there existsa (2 , -subset { β , β , β , β } in { P , P } , i.e., { P , P } hasGRP, yielding a contradiction again.Next, consider the case (ii). Since P and P are reduced simultaneously, there exists a (2 , -subset { β } in { P , P } such that β = β , β . Thus, { β , β , β } is a (2 , -subset in { P , P } . Note, also, that none of the elements β , β , and β is a root of P or P . This comes from two facts: (a) if β isa root of P or P , then reducing P and P via removing β must have reduced P or P , which is a contradiction; and (b)if there exists β ∈ { β , β } such that β is a root of P or P ,then no other element of { β , β , β } belongs, when comparedto β , to more polynomials of degree in { P , . . . , P } (since { β } is a (3 , -subset and there exists no (4 , -subset). Thus, P and P must have been reduced via removing β , whichyields reducing P or P , and hence a contradiction.Since reducing P has reduced P (or P ) and reducing P has reduced P (or P ), then there exist a (2 , -subset { β } in { P , P } and a (2 , -subset { β } in { P , P } such that β = β . Note that β or β is not a root of P or P , respec-tively (otherwise, reducing P (or P ) via removing β (or β ) must have reduced P (or P ), yielding a contradiction).Thus, N = { β , β , β , β } and N = { β , β , β , β } . Sincethere is no (3 , -subset in { P , P , P } , then P has a root β ( = β , β , β , β ) and P has a root β ( = β , β , β , β )such that neither β nor β is a root of P . (Note that β and β may or may not be the same.)Let ˆ P i be the resulting polynomial from P i by removing β , β , β from N i , and let ˆ N i , N i \ { β , β , β } . Let ˆ d i , deg( ˆ P i ) . Note that ( ˆ d , . . . , ˆ d ) = (3 , , , , . Since thereduction set { β , β , β } is acceptable, W ( ˆ P , . . . , ˆ P ) ≡ (by Lemma 1). Thus, { ˆ P , . . . , ˆ P } has GRP (by Theorem 3).This is a contradiction since there exists no (4 , - or (3 , -subset in { ˆ P , . . . , ˆ P } as | ˆ N ∩ ˆ N | = | ˆ N ∩ ˆ N | = 0 , andthere exists no (2 , -subset in { ˆ P , . . . , ˆ P } as | ˆ N ∩ ˆ N | = 2 , | ˆ N ∩ ˆ N | = | ˆ N ∩ ˆ N | = 0 , and | ˆ N ∩ ˆ N |≤ .R EFERENCES[1] M. Yan and A. Sprintson, “Algorithms for Weakly Secure Data Ex-change,” in
Proc. NetCod’13 , Jun. 2013, pp. 1–6.[2] S. H. Dau, W. Song, Z. Dong, and C. Yuen, “Balanced SparsestGenerator Matrices for MDS Codes,” in
Proc. IEEE ISIT’13 , Jul. 2013,pp. 1889–1893.[3] W. Halbawi, T. Ho, and I. Duursma, “Distributed Gabidulin Codes forMultiple-Source Network Error Correction,” in
Proc. NetCod’14 , Jun.2014, pp. 1–6.[4] S. H. Dau, W. Song, and C. Yuen, “On the Existence of MDS Codesover Small Fields with Constrained Generator Matrices,” in
Proc. IEEEISIT’14 , Jun. 2014, pp. 1787–1791.[5] M. Yan, A. Sprintson, and I. Zelenko, “Weakly Secure Data Exchangewith Generalized Reed-Solomon Codes,” in
Proc. IEEE ISIT’14 , Jun.2014, pp. 1366–1370.[6] W. Halbawi, T. Ho, H. Yao, and I. Duursma, “Distributed Reed-SolomonCodes for Simple Multiple Access Networks,” in
Proc. IEEE ISIT’14 ,Jun. 2014, pp. 651–655.[7] W. Halbawi, M. Thill, and B. Hassibi, “Coding with Constraints:Minimum Distance Bounds and Systematic Constructions,” in
Proc.IEEE ISIT’15 , Jun. 2015, pp. 1302–1306.[8] S. H. Dau, W. Song, and C. Yuen, “On Simple Multiple AccessNetworks,”
IEEE Journal on Selected Areas in Communications , vol. 33,no. 2, pp. 236–249, Feb. 2015.[9] W. Halbawi, Z. Liu, and B. Hassibi, “Balanced Reed-Solomon Codes,”in
Proc. IEEE ISIT’16 , Jul. 2016, pp. 935–939.[10] ——, “Balanced Reed-Solomon Codes for All Parameters,” in