On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes
aa r X i v : . [ m a t h . A C ] J a n ON THE PURITY OF RESOLUTIONS OF STANLEY-REISNERRINGS ASSOCIATED TO REED-MULLER CODES
SUDHIR R. GHORPADE AND RATI LUDHANI
Abstract.
Following Johnsen and Verdure (2013), we can associate to anylinear code C an abstract simplicial complex and in turn, a Stanley-Reisner ring R C . The ring R C is a standard graded algebra over a field and its projectivedimension is precisely the dimension of C . Thus R C admits a graded minimalfree resolution and the resulting graded Betti numbers are known to determinethe generalized Hamming weights of C . The question of purity of the minimalfree resolution of R C was considered by Ghorpade and Singh (2020) when C is the generalized Reed-Muller code. They showed that the resolution is purein some cases and it is not pure in many other cases. Here we give a completecharacterization of the purity of graded minimal free resolutions of Stanley-Reisner rings associated to generalized Reed-Muller codes of an arbitrary order. introduction This article concerns a topic that is at the interface of homological aspects ofcommutative algebra and the theory of linear error correcting codes. Our motiva-tion comes from the work of Johnsen and Verdure [10] and the more recent work[7]. In [10], the notion of
Betti numbers of a linear code is introduced. The Bettinumbers of a linear code C of length n are, in fact, the graded Betti numbers of theStanley-Reisner ring R C of the simplicial complex ∆ C on [ n ] := { , . . . , n } whosefaces are precisely the subsets { i , . . . , i t } of [ n ] for which the columns H i , . . . , H i t of a parity check matrix H of C are linearly independent. In [10], it was shownthat the Betti numbers of a linear code determine its generalized Hamming weights.Further, [11, 12] showed that the Betti numbers of a linear code (and its elonga-tions) are also closely related to several classical parameters of that code. Thus it isuseful to know them explicitly. Computation of these Betti numbers is in general,a difficult problem, but it becomes easy, by a formula of Herzog and K¨uhl [9], whenthe corresponding minimal free resolutions are pure. An intrinsic characterizationof purity of the graded minimal free resolutions of Stanley-Reisner rings associatedto arbitrary linear codes was obtained in [7]. As a consequence, known results aboutthe Betti numbers of MDS codes and constant weight codes were easily deduced.One of the most important and widely studied class of linear codes is that ofReed-Muller codes. These codes were introduced by Reed [15] in the binary case andseveral of their properties were established by Muller [14]; see also [4, pp. 20–38].We shall consider Reed-Muller codes in the most general sense, as given by Kasami, Sudhir Ghorpade is partially supported by DST-RCN grant INT/NOR/RCN/ICT/P-03/2018from the Dept. of Science & Technology, Govt. of India, MATRICS grant MTR/2018/000369 fromthe Science & Engg. Research Board, and IRCC award grant 12IRAWD009 from IIT Bombay.Rati Ludhani is supported by Prime Minister’s Research Fellowship PMRF-192002-256 at IITBombay.
Lin and Peterson [13] and by Delsarte, Goethals, and MacWilliams [5]. GeneralizedHamming weights of (generalized) Reed-Muller codes are explicitly known, thanksto the work of Heijnen and Pellikaan [8] (see also [2] and [3]). It is, therefore,natural, to ask for an explicit determination of the Betti numbers of Reed-Mullercodes. The problem would be tractable if we know when the graded minimal freeresolutions of Stanley-Reisner rings of simplicial complexes corresponding to Reed-Muller codes are pure. This question about purity was considered in [7] and ananswer was provided in many, but not all, cases. In this article we build upon thework in [7] and complete it to give a characterization of purity of graded minimalfree resolutions of Stanley-Reisner rings associated to arbitrary Reed-Muller codes.This paper is organized as follows. In Section 2, we review (generalized) Reed-Muller codes and discuss their properties that are relevant for us. Next, in Section 3,the notion of purity of a minimal free resolution is recalled and some key resultsin [7] such as the intrinsic characterization mentioned above and results about thepurity or non-purity of resolutions corresponding to Reed-Muller codes are stated.Our main result on a characterization of purity of free resolutions of Stanley-Reisnerrings associated to Reed-Muller codes is also proved here. As a corollary, we give acharacterization of Reed-Muller codes that are MDS codes.2.
Reed-Muller codes
Standard references for (generalized) Reed-Muller codes are the book of Assmusand Key [1] (especially, Chapter 5) and the seminal paper of Delsarte, Goethals,and MacWilliams [5]. Let us begin by setting some basic notation and terminology.Fix throughout this paper a prime power q and a finite field F q with q elements.Let n, k be integers with 1 ≤ k ≤ n . We write [ n, k ] q -code to mean a q -ary linearcode of length n and dimension k , i.e., a k -dimensional F q -linear subspace of F nq .If the minimum distance of an [ n, k ] q -code is d , then it may be referred to as an[ n, k, d ] q -code. If C is an [ n, k, d ] q -code, then the elements of C of Hamming weight d will be referred to as the minimum weight codewords of C . An [ n, k ] q -code is saidto be nondegenerate if it is not contained in a coordinate hyperplane of F nq . Wedenote by N the set of nonnegative integers.Let m, r be integers such that m ≥ ≤ r ≤ m ( q − V q ( r, m ) := { f ∈ F q [ X , . . . , X m ] : deg( f ) ≤ r and deg X i ( f ) < q for i = 1 , . . . , m } . Note that V q ( r, m ) is a F q -linear subspace of the polynomial ring F q [ X , . . . , X m ].Fix an ordering P , . . . , P q m of the elements of F mq and consider the evaluation mapEv : V q ( r, m ) → F q m q defined by f c f := ( f ( P ) , . . . , f ( P q m )) . (1)Clearly, Ev is a linear map and its image is a nondegenerate linear code of length q m ;this code is called the (generalized) Reed-Muller code of order r , and it is denotedby RM q ( r, m ). The dimension of RM q ( r, m ) is given by the following formulathat can be found in Assmus and Key [1, Theorem 5.4.1]:(2) dim RM q ( r, m ) = r X s =0 m X i =0 ( − i (cid:18) mi (cid:19)(cid:18) s − iq + m − s − iq (cid:19) . In [7, eq. (13)], a somewhat simpler formula for the dimension is stated (withoutproof). It is not difficult to derive it from (2). However, we give an independentand direct proof of the simpler formula below.
URITY OF RESOLUTIONS ASSOCIATED TO REED-MULLER CODES 3
Lemma 1.
Let m, r be integers such that m ≥ and ≤ r ≤ m ( q − . Then (3) dim RM q ( r, m ) = m X i =0 ( − i (cid:18) mi (cid:19)(cid:18) m + r − iqm (cid:19) . Proof.
It is well-known that the map Ev given by (1) is injective. This follows, forinstance, from [6, Lemma 2.1]. Also, if E := { ( v , . . . , v m ) ∈ N m : v + · · · + v m ≤ r } ,then it is easily seen that a basis of V q ( r, m ) is given by B := { X v · · · X v m m : ( v , . . . , v m ) ∈ E and 0 ≤ v j < q for 1 ≤ j ≤ m } . Let E j := { ( v , . . . , v m ) ∈ E : v j ≥ q } for 1 ≤ j ≤ m . The set B is clearly inbijection with E \ ( E ∪· · ·∪ E m ). It is elementary and well-known that | E | = (cid:0) m + rm (cid:1) .By changing v j to v ′ j = v j − q , we also see that | E j | = (cid:0) m + r − qm (cid:1) for 1 ≤ j ≤ m , andmore generally, | E j ∩ · · · ∩ E j i | = (cid:0) m + r − iqm (cid:1) for 1 ≤ j < · · · < j i ≤ m . It followsthat dim RM q ( r, m ) = dim V q ( r, m ) = | B | , and this is equal to | E | − | E ∪ · · · ∪ E m | = (cid:18) m + rm (cid:19) − m X i =1 ( − i − X ≤ j < ··· Remark 2. In case 0 ≤ r < q , formula (3) simplifies to dim RM q ( r, m ) = (cid:0) m + rm (cid:1) .This can also be seen by noting that the set E j in the proof above is empty for each j = 1 , . . . , m when r < q . On the other hand, if r = m ( q − P ν = ( a ν , . . . , a νm ) and consider(4) F ν ( X , . . . , X m ) := m Y j =1 (cid:0) − ( X j − a νj ) q − (cid:1) for ν = 1 , . . . , q m . Note that for any ν ∈ { , . . . , q m } , the polynomial F ν is in V q ( m ( q − , m ) and ithas the property that F ν ( P ν ) = 1 and F ν ( P µ ) = 0 for any µ ∈ { , . . . , q m } with µ = ν . Hence any λ = ( λ , . . . , λ q m ) ∈ F q m q can be written as λ = Ev( F ), where F = λ F + · · · + λ q m F q m . It follows that RM q ( m ( q − , m ) = F q m q . In particular,Lemma 1 yields the following curious identity: m X i =0 ( − i (cid:18) mi (cid:19)(cid:18) ( m − i ) qm (cid:19) = q m or equivalently, m X i =0 ( − i (cid:18) mi (cid:19)(cid:18) iqm (cid:19) = ( − q ) m . It may be interesting to obtain a direct proof of the above identity.We now recall the following important result about the minimum distance andthe minimum weight codewords of Reed-Muller codes. Proposition 3. Let m, r be integers such that m ≥ and ≤ r ≤ m ( q − . Thenthere are unique t, s ∈ N such that (5) r = t ( q − 1) + s and ≤ s ≤ q − . With t, s as above, the minimum distance of RM q ( r, m ) is given by (6) d = ( q − s ) q m − t − . SUDHIR R. GHORPADE AND RATI LUDHANI Further, if f ∈ V q ( r, m ) is given by f ( X , . . . , X m ) = ω t Y i =1 (cid:0) − ( X i − ω i ) q − (cid:1) s Y j =1 ( X t +1 − ω ′ j )(7) where ω , ω , . . . , ω t ∈ F q with ω = 0 and ω ′ , . . . , ω ′ s are any distinct elements of F q , then Ev( f ) is a minimum weight codeword of RM q ( r, m ) . Moreover, everyminimum weight codeword of RM q ( r, m ) is of the form Ev( g ) , where g is obtainedfrom a polynomial of the form (7) by substituting for X , . . . , X t +1 any ( t + 1) linearly independent linear forms in F q [ X , . . . , X m ] .Proof. The formula in (6) follows from [5, Theorem 2.6.1] and [13, Theorem 5]. Theassertion about the minimum weight codewords is proved in [5, Theorem 2.6.3]. (cid:3) We end this section by observing that the Reed-Muller code RM q ( r, m ) is aparticularly nice code when m is small or when r is either very small or very large. Lemma 4. Let m, r be integers such that m ≥ and ≤ r ≤ m ( q − . Then RM q ( r, m ) is an MDS code in each of the following cases: (i) m = 1 , (ii) r = 0 ,(iii) r = m ( q − , and (iv) r = m ( q − − .Proof. (i) if 0 ≤ r < q , then in view of Remark 2 and Proposition 3, we see that RM q ( r, 1) is a [ q, r + 1 , q − r ] q -code, and hence it is an MDS code.(ii) Clearly, RM q (0 , m ) is the 1-dimensional code of length q m spanned by theall-1 vector, and this is evidently an MDS code.(iii) From Remark 2, RM q ( m ( q − , m ) = F q m q , which is obviously an MDS code.(iv) Suppose r = m ( q − − 1. We will show that(8) RM q ( r, m ) = Λ , where Λ := n ( λ , . . . , λ q m ) ∈ F q m q : λ + · · · + λ q m = 0 o . This would imply that RM q ( r, m ) is a [ q m , q m − , q -code, and hence an MDScode. To prove (8), first note that the monomial X q − · · · X q − m is in V q ( m ( q − , m ),but not in the subspace V q ( r, m ). Since we have seen in Remark 2 that Ev gives anisomorphism of V q ( m ( q − , m ) onto F q m q , it follows that dim F q V q ( r, m ) ≤ q m − ⊆ RM q ( r, m ). To this end, we assume withoutloss of generality that the ordering P , . . . , P q m of points of F mq is such that P isthe origin. For 1 ≤ ν ≤ q m , consider the polynomial F ν given by (4), and write F ν = H + G ν , where H := F = m Y j =1 (cid:0) − X q − j (cid:1) and G ν := F ν − H. Note that G ν ∈ V q ( r, m ) for each ν = 1 , . . . , q m . Also, H ( P ) = 1 and H ( P µ ) = 0for 2 ≤ µ ≤ q m . So in view of the properties of F ν noted in Remark 2, we see that G ( P ) = 0 while G ν ( P ) = − G ν ( P ν ) = 1 for 2 ≤ ν ≤ q m , and moreover, G ν ( P µ ) = 0 for 2 ≤ ν, µ ≤ q m with ν = µ . Thus given any λ = ( λ , . . . , λ q m ) ∈ Λ,the polynomial G := P q m λ =1 λ ν G ν ∈ V q ( r, m ) and Ev( G ) = λ. This proves (8). (cid:3) Remark 5. In [7, pp. 8–9], the results in Lemma 4, especially (iv), were deducedby appealing to the structure of duals of Reed-Muller codes. Here we have chosento give a more direct and elementary proof. We remark also that the converse of theresult in Lemma 4 is true. An indirect proof of this is given later; see Corollary 11. URITY OF RESOLUTIONS ASSOCIATED TO REED-MULLER CODES 5 Characterizations of Purity Let n, k ∈ N with 1 ≤ k ≤ n and let C be an [ n, k ] q -code. We have explainedin the Introduction how one can associate an abstract simplicial complex ∆ C to C . Note that this complex is independent of the choice of a parity check matrixof C . Let R := F q [ x , . . . , x n ] denote the polynomial ring in n variables over F q and let I C denote the ideal of R generated by the monomials x i · · · x i t where { i , . . . , i t } vary over non-faces, i.e., over subsets of [ n ] := { , . . . , n } that are not in∆ C . The Stanley-Reisner ring R C corresponding to ∆ C (with the base field F q ) is,by definition, the quotient R/I C . We call R C the Stanley-Reisner ring associatedto C . Clearly, R C is a standard graded F q -algebra and as noted in [7, § R C isCohen-Macaulay and it admits an N -graded minimal free resolution of the form(9) F k −→ F k − −→ · · · −→ F −→ F −→ R ∆ −→ F = R and each F i is a graded free R -module of the form(10) F i = M j ∈ Z R ( − j ) β i,j for i = 0 , , . . . , k. The nonnegative integers β i,j thus obtained are called the Betti numbers of C . Theresolution (9) is said to be pure of type ( d , d , . . . , d k ) if for each i = 0 , , . . . , k ,the Betti number β i,j is nonzero if and only if j = d i . If, in addition, d , . . . , d k are consecutive, then the resolution is said to be linear . We remark that the Bettinumbers β i,j as well as the properties of purity and linearity depend only on C andthey are independent of the choice of a minimal free resolution of R C .The result below is due to Johnsen and Verdure [10]; see also [7, Corollary 3.9]. Proposition 6. Let C be an [ n, k ] q -code. Then C is an MDS code if and only if C is nondegenerate and every N -graded minimal free resolution of R C is linear. We will now recall the intrinsic characterization of purity given in [7] and alludedto in the Introduction. But first, we review some relevant terminology about codes.Let n, k and C be as above. By a subcode of C we mean a F q -linear subspace of C . Given a subcode D of C , the support of D and the weight of D are defined bySupp( D ) := { i ∈ [ n ] : ∃ ( c , . . . , c n ) ∈ D with c i = 0 } and wt( D ) := | Supp( D ) | . Given any c ∈ C , we often denote by Supp( c ) and wt( c ) the support of h c i and theweight of h c i , respectively, where h c i denotes the subcode of C spanned by c . For1 ≤ i ≤ k , the i th generalized Hamming weight of C is defined by d i ( C ) := min { wt( D ) : D a subcode of C with dim D = i } . It is well-known that d ( C ) is the minimum distance of C and d i ( C ) < d i +1 ( C ) for1 ≤ i < k . Note that C is nondegenerate if and only if d k ( C ) = n . An i -dimensionalsubcode D of C is said to be i -minimal if its support is minimal among the supportsof all i -dimensional subcodes of C , i.e., Supp( D ′ ) * Supp( D ) for any i -dimensionalsubcode D ′ of C , with D ′ = D .We are now ready to state (an equivalent version of) the intrinsic characterizationof purity given in [7, Theorem 3.6]. Proposition 7. Let C be an [ n, k ] q -code and let d < · · · < d k be its generalizedHamming weights. Also, let R C be the Stanley-Reisner ring associated to C . Thenevery N -graded minimal free resolution of R C is not pure if and only if there existsan i ∈ { , . . . , k } and an i -minimal subcode D i of C such that wt( D i ) > d i . SUDHIR R. GHORPADE AND RATI LUDHANI We summarize below the results in [7] about the purity and non-purity of gradedminimal free resolutions of Stanley-Reisner ring associated to Reed-Muller codes. Proposition 8. Let m, r be integers such that m ≥ and ≤ r ≤ m ( q − . Also,let t, s be unique nonnegative integers satisfying (5) . Then every N -graded minimalfree resolution of the Stanley-Reisner ring associated to R M q ( r, m ) is (i) pure if r = 1 , (ii) not pure if q = 2 , m ≥ , and < r ≤ m − , and (iii) not pure if m ≥ , < r < m ( q − − , and s = 1 .Proof. The assertion in (i) is proved in [7, Theorem 4.1], while the assertions in (ii)and (iii) are proved in [7, Proposition 4.4] and [7, Theorem 4.11], respectively. (cid:3) The values of q, m, r not covered by (i)–(iv) in Lemma 4 and (i)–(iii) in Proposi-tion 8 are precisely q ≥ m ≥ 2, and r = q, q − , . . . , ( m − q − ( m − m − q − ( m − 2) is excluded if q = 3. This is taken care of by the following. Lemma 9. Let m, r be integers such that m ≥ and < r < m ( q − − . Alsolet t, s be unique integers satisfying (5) . Assume that q ≥ and also that s = 1 .Then every N -graded minimal free resolution of the Stanley-Reisner ring associatedto the Reed-Muller code R M q ( r, m ) is not pure.Proof. The conditions on m, r and our assumptions imply that 1 ≤ t ≤ m − q = 3, then 1 ≤ t ≤ m − 2. Also note that by Proposition 3, theminimum distance of R M q ( r, m ) is given by d = ( q − q m − t − . We will divide theproof in two cases according as q > q = 3. Case 1. q > F q = { ω , . . . , ω q } , and let ω ′ , ω ′ be two distinct elements of F q . Define Q ( X , . . . , X m ) := t − Y i =1 ( X q − i − ! q Y j =3 ( X t − ω j ) Y k =1 ( X t +1 − ω ′ k ) ! . Then deg( Q ) = ( t − q − 1) + ( q − 2) + 2 = ( t − q − 1) + q = t ( q − 1) + 1 = r ,and thus Q ∈ V q ( r, m ). For i = 1 , 2, let A i := (cid:8) a = ( a , . . . , a m ) ∈ F mq : a = · · · = a t − = 0 , a t = ω i and a t +1 6∈ { ω ′ , ω ′ } (cid:9) . Then Supp( c Q ) = A ∪ A . Observe that A and A are disjoint. Consequently,wt( c Q ) = 2( q − q m − t − and therefore wt( c Q ) > d = ( q − q m − t − , where the last inequality follows since q > 3. Thus c Q is not a minimum weightcodeword. We claim that the 1-dimensional subcode h c Q i is 1-minimal. This claimtogether with Proposition 7 would imply the desired result. To prove the claim, as-sume the contrary. Thus, suppose there is F ∈ V q ( r, m ), such that c F is a minimumweight codeword of RM q ( r, m ) and Supp( c F ) ( Supp( c Q ). By Proposition 3, F must be of the form(11) F ( X , . . . , X m ) = ω t Y i =1 (1 − L q − i ) ! ( L t +1 − ω )for some linearly independent linear polynomials L , . . . , L t +1 in F q [ X , . . . , X m ]and some ω , ω ∈ F q with ω = 0. Note that Supp( c F ) = A ′ , where(12) A ′ := (cid:8) a = ( a , . . . , a m ) ∈ F mq : L i ( a ) = 0 for 1 ≤ i ≤ t and L t +1 ( a ) = ω (cid:9) . URITY OF RESOLUTIONS ASSOCIATED TO REED-MULLER CODES 7 Since Supp( c F ) ⊂ Supp( c Q ), we obtain A ′ ⊂ A ∪ A . We now assert that A ′ is disjoint from one of the A i . Indeed, if the assertion were not true, then wecan choose P i ∈ A ′ ∩ A i for i = 1 , 2. Write b i := L t +1 ( P i ) for i = 1 , 2. Since P i ∈ A ′ , we see that b i = ω for i = 1 , 2. Now pick λ ∈ F q such that λ = 0 , − λ ) b + λb = ω , which is possible because q ≥ Define P λ := (1 − λ ) P + λP .Then P λ ∈ A ′ , and this contradicts the inclusion A ′ ⊂ A ∪ A because the t th coordinate of P λ is neither ω nor ω . This proves the above assertion. ThusSupp( c F ) = A ′ ⊆ A i for some i . But then ( q − q m − t − ≤ ( q − q m − t − , which isa contradiction. This proves the claim and hence the desired result when q > Case 2. q = 3.In this case 1 ≤ t ≤ m − 2, as noted earlier. Write F q = { ω , ω , ω } . Define Q ( X , . . . , X m ) := (cid:18) t − Y i =1 ( X q − i − (cid:19) ( X t − ω )( X t +1 − ω )( X t +2 − ω ) . Then deg( Q ) = ( t − q − t ( q − r , since q = 3, and so Q ∈ V q ( r, m ).Let E := (cid:8) a = ( a , . . . , a m ) ∈ F mq : a = · · · = a t − = 0 (cid:9) , and for i = 1 , 2, let A i := { a = ( a , . . . , a m ) ∈ E : a t = ω i and a t +1 , a t +2 ∈ { ω , ω }} ,A ′ i := { a = ( a , . . . , a m ) ∈ E : a t +1 = ω i and a t , a t +2 ∈ { ω , ω }} , and A ′′ i := { a = ( a , . . . , a m ) ∈ E : a t +2 = ω i and a t , a t +1 ∈ { ω , ω }} . Then Supp( c Q ) = A ∪ A = A ′ ∪ A ′ = A ′′ ∪ A ′′ and wt( c Q ) = 2 q m − t − . Note thatwt( c Q ) > ( q − q m − t − , since q = 3. Thus, as in Case 1, it suffices to show thatthere does not exist any F ∈ V q ( r, m ) such that c F is a minimum weight codewordand Supp( c F ) ( Supp( c Q ). Suppose, if possible, there is such F . Then it mustbe of the form (11), and its support is given by the set A ′ in (12). Now write F q \ { ω } = { u , u } , and for i = 1 , 2, let B i := (cid:8) a = ( a , . . . , a m ) ∈ F mq : L i ( a ) = 0 for 1 ≤ i ≤ t and L t +1 ( a ) = u i (cid:9) . Note that each B i is an affine space (i.e., a translate of a linear subspace) in F mq and Supp( c F ) = B ∪ B . Thus B ∪ B ⊂ A ∪ A . We claim that B ⊆ A i forsome i ∈ { , } . Indeed, if this were not true, then we can find P i ∈ B ∩ A i foreach i = 1 , 2. Since q = 3, we can choose λ ∈ F q such that λ = 0 , 1. Consider P λ := (1 − λ ) P + λP . Since B is an affine space, P λ ∈ B . On the other hand,the t th coordinate of P λ is neither ω nor ω , and hence P λ A ∪ A . Thiscontradicts the inclusion B ⊂ A ∪ A , and so the Claim is proved. In a similarmanner, we see that B ⊆ A ′ j and B ⊆ A ′′ k for some j, k ∈ { , } . It follows that B ⊆ A i ∩ A ′ j ∩ A ′′ k . But clearly, | B | = q m − t − and | A i ∩ A ′ j ∩ A ′′ k | = q m − t − . So weobtain q m − t − ≤ q m − t − , which is a contradiction. This completes the proof. (cid:3) We are now ready to prove the main result of this article. Theorem 10. Let m, r ∈ N be such that m ≥ and ≤ r ≤ m ( q − . Then every N -graded minimal free resolution of the Stanley-Reisner ring associated to the Reed-Muller code RM q ( r, m ) is pure if and only if m = 1 or r ≤ or r ≥ m ( q − − .Proof. Follows from Lemma 4, Proposition 6, Proposition 8, and Lemma 9. (cid:3) If b = b , then the only condition on λ is that λ = 0 , 1, whereas if b = b , then it suffices tochoose λ ∈ F q such that λ = 0 , λ = ( ω − b ) / ( b − b ). s an application, we show that the converse of the result in Lemma 4 is true. Corollary 11. Let m, r ∈ N be such that m ≥ and ≤ r ≤ m ( q − . Then theReed-Muller code RM q ( r, m ) is an MDS code if and only if m = 1 or r = 0 or r ≥ m ( q − − .Proof. If m = 1 or r = 0 or r ≥ m ( q − − 1, then by Lemma 4, RM q ( r, m )is an MDS code. Conversely, suppose RM q ( r, m ) is an MDS code. Then byProposition 6, every N -graded minimal free resolution of its Stanley-Reisner ringis pure. So by Theorem 10, we must have m = 1 or r ≤ r ≥ m ( q − − m ≥ 2, then the case r = 1 is ruled out because by [7, Theorem 4.1], thegeneralized Hamming weights (which coincide with the “shifts” in the resolution)of RM q (1 , m ) are given by d i = q m − ⌊ q m − i ⌋ for 1 ≤ i ≤ m + 1, and these areclearly non-consecutive if m ≥ 2, and so by Proposition 6, RM q (1 , m ) cannot bean MDS code if m ≥ 2. Thus we must have m = 1 or r = 0 or r ≥ m ( q − − (cid:3) References [1] E. F. Assmus Jr. and J. D. Key, Designs and their Codes , Cambridge Tracts in Math. , , Cambridge Univ. Press, 1992.[2] P. Beelen and M. Datta, Generalized Hamming weights of affine cartesian codes, FiniteFields Appl. (2018), 130–145.[3] P. Beelen, A note on the generalized Hamming weights of Reed-Muller codes, Appl.Algebra Engrg. Comm. Comput. (2019), 233–242.[4] E. R. 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