In this section, we introduce a class of objects known as accounts, which will simplify the expression and proof of our results. IfY is a set, we define anaccount onY to be a function from Y into Z. If µis an account on Y and y∈Y, we use the notationµy for the value of µ at y. We say µ has k instances of y to mean µy =k. The set of accounts on Y, when equipped with addition, forms an Abelian group that we denote byZ[Y], and we sometimes write and manipulate an account µas if it were the formal sum P
y∈Y µyy. The multisets with elements from Y are regarded in a natural way as the accounts on Y that take only nonnegative values. The subsets of Y are regarded as the accounts on Y that take only values 0 and 1. The set of multisets with elements from Y is denotedN[Y].
Suppose that Y1 ⊆ Y2. Then we regard Z[Y1] as a subset of Z[Y2] by regarding each µ:Y1 →Z as the function from Y2 toZ that vanishes on Y2rY1 and extends the original functionµ. In this way, we also have N[Y1]⊆N[Y2]. Conversely, ifµ∈Z[Y2] is supported on Y1, then we can regard µ as an element of Z[Y1] via restriction of domain. Indeed, we often indicate that µ∈Z[Y2] is supported onY1 by writingµ∈Z[Y1].
Thesizeof an accountµonY isP
y∈Y µy and is denoted|µ|. This is just the cardinality of the account if the account is a set or a multiset. If µis a multiset, we use the notation µ! as a shorthand for Q
y∈Y µy!. Thus there are|µ|!/µ! distinct ways of arranging the |µ|
elements of the multiset µ into an ordered |µ|-tuple. Multisets will be easier to use than the sequences that are employed in the definitions of the parameters `(C), `mc(C), `ss(C), and`ssmc(C) that appear in Section 1.1 of the Introduction. Multisets are more natural than sequences since the order of terms in these sequences is irrelevant.
Since we shall often be dealing with accounts on finite Cartesian products of sets, we establish some conventions for dealing with accounts of k-tuples. Ifk is a positive integer, and µ ∈Z[B1× · · · ×Bk], then we write µb1,...,bk instead of µ(b1,...,bk) for the value of µat (b1, b2, . . . , bk) ∈ B1 ×B2 × · · · ×Bk. If 1 ≤ j1 < j2 < · · · < js ≤ k, then we define the projection ofµtoBj1×· · ·×Bjs, denoted prBj
1×···×Bjsµ, to be the account onBj1×· · ·×Bjs with
prBj
1×···×Bjsµ
bj1,...,bjs = X
(c1,...,ck)∈B1×···×Bk cj1=bj1,...,cjs=bjs
µc1,...,ck
for each bj1, . . . , bjs ∈Bj1× · · · ×Bjs. Note that
prBj
1×···×Bjs(µ1+µ2) = prBj
1×···×Bjsµ1+ prBj
1×···×Bjsµ2.
For example, if µ ∈ Z[V ×W], then prW µ ∈ Z[W] with (prWµ)w = P
v∈V µv,w for all w∈W.
Suppose thatk∈N,j∈ {1,2, . . . , k−1},µ∈Z[B1× · · · ×Bk], andb1 ∈B1, . . . , bj ∈Bj. Then we define µb1,...,bj to be the account in Z[Bj+1× · · · ×Bk] with (µb1,...,bj)bj+1,...,bk = µb1,...,bj,bj+1,...,bk for all bj+1∈Bj+1, . . . , bk ∈Bk. For example, if µ∈Z[V ×W] and v∈V, thenµv ∈Z[W] with (µv)w =µv,w for allw∈W.
Throughout this thesis, we letH ={0,1, . . . , e−1}. Ifµ∈Z[H], we define thep-weighted summation of µ, denoted Σµ, by
Σµ= X
h∈H
µhph (modq−1). (2.3)
Note that Σ is a homomorphism from Z[H] into the group Z/(q −1)Z under addition.
If µ ∈ Z[H] with Σµ = 0, we call µ a Delsarte-McEliece account; the Delsarte-McEliece accounts form a subgroup ofZ[H]. Recall the modular condition used in the definition of the parameterωmc(C) in Section 1.1 of the Introduction. There we wanted to find the minimum length of unity-product sequences of the form ap1j1, ap2j2, . . . , apnjn, where each ai is in some q-closed subset S of A and each ji ∈ N, subject to the conditionpj1 +pj2 +· · ·+pjn ≡0 (modq−1). It harms nothing to further stipulate that eachji lie in H, for S is q-closed and the congruence is modulo q−1. Then the modular condition is equivalent to saying that the multiset µ ∈ N[H] with elements j1, . . . , jn is Delsarte-McEliece. This modular condition was discovered by Delsarte and McEliece [18], hence we attach their names to accounts in Z[H] that correspond to it. The following fact is very useful and not difficult to prove:
Lemma 2.15. A Delsarte-McEliece account inZ[H]has size divisible by p−1. The unique nonempty Delsarte-McEliece multiset inN[H]of minimal cardinality hasp−1instances of each element of H, and thus has a cardinality of e(p−1).
Proof. A very similar thing is proved in Lemma 2.1 of [61]. If µ ∈ Z[H] is a Delsarte- McEliece account, reduce (2.3) modulo p−1 to obtain 0 ≡ P
h∈Hµh (modp−1). If we assume that µ is a nonempty Delsarte-McEliece multiset of minimal cardinality, then we claim that µh < p for allh ∈H. Otherwise, we could make a smaller nonempty Delsarte- McEliece multiset µ0 by removing p copies of an element h and adding one copy of the element h+ 1 (where we treat h+ 1 as 0 if h =e−1). So 0 ≤µh ≤ p−1 for all h ∈ H and not all µh are zero. If we had µh < p−1 for any h, then 0 < P
h∈Hµhph < q−1, contradicting the fact thatµ is Delsarte-McEliece. So µh =p−1 for allh∈H.
Ifµ∈N[H] andr ∈Zp[ζq0−1] or GR(pd, ee0), we define Frµ(r) =Q
h∈H Frh(r)µh
. Note that Frµ(rs) = Frµ(r)Frµ(s) and that Frµ1+µ2(r) = Frµ1(r)Frµ2(r).
We shall often work with accounts of elements inA. We use multisets inN[A] instead of the sequences of elements of A used to define the parameters ω(C) and `(C) in Section 1.1 of the Introduction. This is more natural, since the order of the sequences was irrelevant
there. If λ∈Z[A], then we define theproduct ofλ, denoted Πλ, to be
Πλ= Y
a∈A
aλa.
Note that Π is a homomorphism fromZ[A] (under addition) into the groupA. If Πλ= 1A, we say that λis aunity-product account. Ifλ⊆Z[{1A}], then it is trivially unity-product, and we say that λ is all-unity; otherwise λis not all-unity. If f is a function from A into GR(pd, ee0) or Zp[ζq0−1] (for example, f might be the Fourier transform of an element of GR(pd, e)[A] orZp[ζq−1][A]), then we define the evaluation of f at λ, denoted f(λ), by
f(λ) = Πa∈Af(a)λa.
Note that if F:A→ Zp[ζq0−1] andf =π◦F, then π(F(λ)) =f(λ) for all λ∈Z[A]. Also note that f(λ1+λ2) =f(λ1)f(λ2).
Suppose for the rest of this section that I is a finite set. We shall often need to work with accounts on sets likeI×H,I×A, andI×H×A. Multisets inN[I×A] will replace the sequences used to define the parameters ωss(C), `ss(C), ωssmc(C), and `ssmc(C) in Section 1.1 of the Introduction. If λ∈Z[I×A], then we define theproductofλ, denoted Πλ, to be
Πλ= Y
(i,a)∈I×A
aλi,a.
Note that Π is a homomorphism from Z[I ×A] (under addition) into the group A. If Πλ= 1A, we say that λis a unity-product account. If λ⊆Z[I × {1A}], or, equivalently, if prAλ∈N[{1A}], then λis trivially unity-product, and we say thatλisall-unity; otherwise λisnot all-unity.
We also shall work with accounts of elements in H ×A. When e > 1, multisets in N[H×A] will replace the sequences used to define the parameters ωmc(C) and `mc(C) in Section 1.1 of the Introduction. If λ∈Z[H×A], then we define theproduct of λ, denoted Πλ, to be
Πλ= Y
(h,a)∈H×A
aphλh,a.
Note that Π is a homomorphism from Z[H ×A] (under addition) into the group A. If Πλ= 1A, we say that λis aunity-product account. Ifλ⊆Z[H× {1A}], or, equivalently, if prAλ∈Z[{1A}], then λis trivially unity-product, and we say thatλisall-unity; otherwise λ is not all-unity. If f is a function from A into GR(pd, ee0) or Zp[ζq0−1] (for example, f might be the Fourier transform of an element of GR(pd, e)[A] orZp[ζq−1][A]), then we define theevaluation of f atλ, denoted f(λ), by
f(λ) = Π(h,a)∈H×AFrh(f(a))λh,a.
Note that if F: A → Zp[ζq0−1] and f = π ◦F, then π(F(λ)) = f(λ) for all λ ∈ Z[A], because π commutes with Fr. Also note that f(λ1+λ2) =f(λ1)f(λ2). If λ ∈Z[H×A], then prHλ∈Z[H]. We say that λis Delsarte-McEliece if and only if prHλ is a Delsarte- McEliece account in Z[H], as defined above. Recall the modular condition used in the definition of the parameterωmc(C) in Section 1.1 of the Introduction. Supposeλis a multiset in N[H×A] and suppose that its elements are (h1, a1),(h2, a2), . . . ,(hn, an), listed with multiplicity (but order is unimportant). Suppose we form the sequenceap1h1, ap2h2, . . . , apnhn. Then this sequence meets the modular condition ph1 +ph2 +· · ·+phn ≡0 (modq−1) of Section 1.1 if and only if Σ prHλ= 0, i.e., if and only ifλis Delsarte-McEliece. Concerning the possible sizes of Delsarte-McEliece accounts in Z[H × A], Lemma 2.15 implies the following:
Lemma 2.16. A Delsarte-McEliece account in Z[H ×A] has size divisible by p−1. A nonempty Delsarte-McEliece multiset in N[H×A] has cardinality at leaste(p−1).
Proof. If λ∈Z[H×A] is Delsarte-McEliece, then prHλ∈N[H] is Delsarte-McEliece and has the same size. Then apply Lemma 2.15.
We shall need to consider accounts of elements in sets of the form I×H×A. Multisets in N[I×H×A] will replace the sequences used to define the parameters ωmc(C1, . . . ,Ct) and `mc(C1, . . . ,Ct) in Section 1.1 of the Introduction. Ifλ∈Z[I×H×A], then we define
theproduct ofλ, denoted Πλ, to be
Πλ= Y
(i,h,a)∈I×H×A
aphλi,h,a.
Note that Π is a homomorphism fromZ[I×H×A] (under addition) into the groupA. If Πλ= 1A, we say thatλis aunity-productaccount. Ifλ⊆Z[I×H× {1A}], or, equivalently, if prAλ∈Z[{1A}], thenλis trivially unity-product, and we say thatλisall-unity; otherwise λisnot all-unity.