Note that we never used R in the definition of this sequence λ(0), . . . , λ(m), so the final multiset in the sequence is independent ofR.
This lemma shows us that ifλ∈N[H×A], then there is a unique reducedκ∈N[H×A]
such that CoR(κ) = CoR(λ) for any set R of q-class representatives of A. Using the same terminology as we did for multisets inN[A], we call thisκthe reduction ofλand denote it by Red(λ). Then we obtain results analogous to Corollaries 2.22 and 2.23.
Corollary 2.26. Suppose that λ1, λ2 ∈N[H×A] and R is a set of q-class representatives of A. Then Red(λ1) = Red(λ2) if and only if CoR(λ1) = CoR(λ2).
In view of Lemma 2.20, we have the following result:
Corollary 2.27. Suppose that λ ∈ N[H×A] and R is a set of q-class representatives of A. Suppose that c ∈ GR(pd, e)[A] such that ˜c(a) is zero or a power of π(ζq0−1) for all a ∈A. Let C be the element in Zp[ζq0−1][A] with C˜ =τ ◦˜c. Then c(λ) = ˜˜ c(Red(λ)) and C(λ) = ˜˜ C(Red(λ)).
We transport the notion of reduction to elements ofN[I×H×A], whereIis some finite set. Thereductionofλ∈N[I×H×A] is the elementκ∈N[I×H×A] withκi = Red(λi) for all i∈I. Then we obtain an analogue of Corollary 2.24 above.
Corollary 2.28. Let I be a finite set. For each i ∈ I, let Si be a q-closed subset of A, and suppose that λ ∈ N[I×H×A] with λi ∈ N[H ×Si] for each i ∈ I. Then for each i ∈ I, we have (Red(λ))i ∈ N[H×Si]. For each i ∈ I, (Red(λ))i is Delsarte-McEliece if and only if λi is Delsarte-McEliece. Also Π Red(λ) = Πλ and Red(λ) is all-unity if and only ifλ is all-unity. Furthermore, for each i∈I, (Red(λ))i =∅ if and only if λi =∅, and
|(Red(λ))i|=|λi| −ki(p−1) for some nonnegative integer ki. If λ is not reduced, at least one of theseki is strictly positive. IfR is a set of p-class representatives ofA, thenRed(λ) has CoR([Red(λ)]i) = CoR(λi) for each i∈I.
automorphism Fr. This is done to prove that such terms, known to be in Zp[ζq0−1], are actually elements ofZp (which is comforting, since they are supposed to bep-adic estimates of elements of Z). Readers should probably ignore this section until they encounter the Frobenius action in the proofs of the aforementioned theorems. The development here is repetitious, as in the previous section, because we need several variants of the same idea.
The proofs are routine, but we include them with a fair amount of detail for the record.
When e= 1, i.e., whenq =p, we introduce the Frobenius action, denoted FrA, on the groupA. We let FrA(a) =ap. Note that FrAe0(a) =aq0 =a, by our choice ofe0 andq0. Thus FrAis a permutation ofA. Also note that the orbits of FrAare thep-classes. We extend FrA
to act on elements of Z[A] by sending P
a∈Aλaa to P
a∈AλaFrA(a) = P
a∈Aλaap. Note that FrAe0(λ) =λ for any λ∈ Z[A], so FrA is a permutation of Z[A]. Also note that FrA
preserves size. There are many other such useful properties that we summarize here:
Lemma 2.29. If e = 1, then FrA is a permutation of Z[A] with FrAe0 the identity. FrA
preserves size. FrA takes multisets to multisets and (FrA(λ))! = λ! for all λ ∈ N[A]. If S is a p-closed set and λ∈ Z[S], then FrA(λ) ∈ Z[S]. Additionally, FrA(λ) is all-unity if and only if λ is all-unity. Further, Π FrA(λ) = (Πλ)p, so that FrA(λ) is unity-product if and only if λ is unity-product. If c∈ Z/pdZ[A] and C is the element of Zp[ζq0−1][A] with C˜=τ◦c, and if˜ λ∈Z[A], then C(Fr˜ A(λ)) = Fr( ˜C(λ)).
Proof. It has already been noted that FrA is a size-preserving permutation with FrAe0 the identity. Since FrA permutes the elements ofA, if we list the|A|coefficients in the formal sumλ=P
a∈Aλaaand then list the|A|coefficients in the formal sum FrA(λ) =P
a∈Aλaap, we get the same list modulo ordering. Thusλtakes multisets to multisets and FrA(λ)! =λ!.
Since the orbits of the action of FrA on A are p-classes, FrA always takes an element of a p-closed set S into the same set S. So FrA takes Z[S] into itself. In particular, FrA maps all-unity accounts to all-unity accounts, and furthermore, if FrA(λ) is all-unity, then λ = FrAe0(λ) is all-unity. Note that Π FrA(λ) = Q
a∈Aa(FrA(λ))a = Q
a∈A(ap)(FrA(λ))ap = Q
a∈A(ap)λa = (Πλ)p.
Finally, suppose that c∈Z/pdZ[A] and C is the element ofZp[ζq0−1][A] with ˜C=τ◦c.˜ Then by Lemma 2.9, we have C ∈Zp[A]. Now suppose that λ∈Z[A]. Then ˜C(FrA(λ)) =
Q
a∈AC(a)˜ (FrA(λ))a =Q
a∈AC(a˜ p)(FrA(λ))ap =Q
a∈AC(a˜ p)λa.Now use Proposition 2.8 (rec- ognizing thatq=psincee= 1) to obtain ˜C(FrA(λ)) =Q
a∈AFr( ˜C(a))λa = Fr( ˜C(λ)).
IfI is a finite set, we extend the Frobenius action FrAtoZ[I×A] so that ifλ∈Z[I×A], (FrA(λ))i = FrA(λi). We can easily deduce what we need to know about this action on Z[I ×A] from the above lemma.
Corollary 2.30. If e= 1, then FrA is a permutation of Z[I×A] with FrAe0 the identity.
FrA preserves size. Furthermore, ifλ∈Z[I×A], then |(FrA(λ))i|=|λi|for all i∈I. Thus prIFrA(λ) = prIλ. FrA takes multisets to multisets and(FrA(λ))! =λ!for allλ∈N[A]. If Si is a p-closed set for each i∈ I, and if λi ∈Z[Si] for each i∈I, then (FrA(λ))i ∈Z[Si] for each i ∈ I. Additionally, FrA(λ) is all-unity if and only if λ is all-unity. Further, Π FrA(λ) = (Πλ)p, so thatFrA(λ) is unity-product if and only ifλis unity-product. Suppose thatci ∈Z/pdZ[A]for eachi∈I and that Ci is the element ofZp[ζq0−1][A]withC˜i=τ◦˜ci for each i∈I. If λ∈Z[I×A], then Q
i∈IC˜i([FrA(λ)]i) = Fr Q
i∈IC˜i(λ) .
For any value of e, we also define the Frobenius action FrA as a function from the set H×A to itself, wherein
FrA(h, a) =
(h+ 1, a) ifh < e−1, (0, aq) ifh=e−1.
Note that FrAe(h, a) = (h, aq), and so FrAee0(h, a) = (h, aq0) = (h, a), by our choice of e0 and q0. Thus FrA is a permutation ofH×A. Also note that the orbits of FrA are sets of the form H×B, whereB is a q-class in A. We extend FrA to act on elements ofZ[H×A]
by sendingP
(h,a)∈H×Aλh,a(h, a) toP
(h,a)∈H×Aλh,aFrA(h, a). Note that FrAee0(λ) =λfor any λ ∈ Z[A], so FrA is a permutation of Z[H×A]. Also note that FrA preserves size.
There are many other such useful properties, analogous to those in Lemma 2.29, which we summarize here:
Lemma 2.31. FrA is a permutation of Z[H×A] with FrAee0 the identity. FrA preserves size. FrA takes multisets to multisets and (FrA(λ))! = λ! for all λ ∈ N[H×A]. If S is a q-closed set and λ∈ Z[H×S], then FrA(λ)∈ Z[H×S]. Additionally, FrA(λ) is all-unity
if and only if λ is all-unity. Further, Π FrA(λ) = (Πλ)p, so that FrA(λ) is unity-product if and only if λ is unity-product. We have Σ prH(FrA(λ)) =p(Σ prHλ), so that FrA(λ) is Delsarte-McEliece if and only if λ is Delsarte-McEliece. If c ∈ GR(pd, e)[A] and C is the element of Zp[ζq0−1][A]with C˜=τ◦c, and if˜ λ∈Z[H×A], then C(Fr˜ A(λ)) = Fr( ˜C(λ)).
Proof. It has already been noted above that FrAis a size-preserving permutation with FrAee0
the identity. Since FrA permutes the elements of H×A, if we list the e|A| coefficients in the formal sum λ=P
(h,a)∈H×Aλh,a(h, a) and then list the e|A|coefficients in the formal sum FrA(λ) = P
(h,a)∈H×Aλh,aFrA(h, a), we get the same list modulo ordering. Thus λ takes multisets to multisets and FrA(λ)! =λ!. Recall that the orbits of the action of FrAon H×A are Cartesian products ofH withq-classes. Thus, if S is q-closed, then FrA always takes an element of H×S into the same set H ×S. So FrA takes N[H×S] into itself.
Therefore FrA takes all-unity accounts to all-unity accounts, and furthermore, if FrA(λ) is all-unity, then λ= FrAee0(λ) is all-unity.
We now show that Π FrA(λ) = (Πλ)p. First we verify this for the account that is a singleton set {(h, a)}. We write this account as the formal sum 1(h, a), or just (h, a).
Indeed, if h < e −1, we have Π FrA(h, a) = Π(h + 1, a) = aph+1 = (Π(h, a))p. On the other hand, if h = e− 1, then Π FrA(e −1, a) = Π(0, aq) = aq =
ape−1p
= (Π(e−1, a))p. Now we consider an arbitrary λ ∈ Z[H ×A]. By the properties of Π, we can calculate Π FrA(λ) = Π
P
(h,a)∈H×Aλh,aFrA(h, a)
=Q
(h,a)∈H×A(Π FrA(h, a))λh,a = Q
(h,a)∈H×A(Π(h, a))pλh,a =h Π
P
(h,a)∈H×Aλh,a(h, a)ip
= (Πλ)p.
Next, we show that Σ prHFrA(λ) = p(Σ prHλ). First we check this for the sin- gleton set (h, a) (we are representing it as a formal sum, since it is an account). In- deed, if h < e−1, then Σ prHFrA((h, a)) = Σ prH(h+ 1, a) = ph+1 = pΣ prH(h, a). If h = e−1, recall that Σ : Z[H] → Z/(q −1)Z, and then check Σ prHFrA((e−1, a)) = Σ prH(0, aq) = 1 = q = p(pe−1) = pΣ prH(e−1, a) in Z/(q −1)Z. Then, for an ar- bitrary λ ∈ Z[H × A], we have Σ prHFrA(λ) = Σ prH
P
(h,a)∈H×Aλh,aFrA(h, a)
= P
(h,a)∈H×Aλh,a(Σ prHFrA(h, a)) = P
(h,a)∈H×Aλh,ap(Σ prH(h, a)). Therefore, we have Σ prHFrA(λ) = pΣ prH
P
(h,a)∈H×Aλh,a(h, a)
= pΣ prHλ. Since p is coprime to q−1, this means that Σ prHλ= 0 if and only if Σ prHFrA(λ) = 0, i.e.,λis Delsarte-McEliece if
and only if FrA(λ) is Delsarte-McEliece.
Finally, suppose thatc∈GR(pd, e)[A] andCis the element ofZp[ζq0−1][A] with ˜C=τ◦˜c.
Then by Lemma 2.9, we haveC ∈Zp[ζq−1][A]. Now suppose thatλ∈Z[H×A]. We want to show that ˜C(FrA(λ)) = Fr( ˜C(λ)). First we verify this when λis the singleton set (h, a).
Indeed, ifh < e−1, we have ˜C(FrA((h, a))) = ˜C((h+1, a)) = Frh+1( ˜C(a)) = Fr
C((h, a))˜ . On the other hand, if h = e−1, then we have ˜C(FrA((e−1, a))) = ˜C((0, aq)) = ˜C(aq).
But now use Proposition 2.8 to get ˜C(aq) = Fre( ˜C(a)) = Fr
C((e˜ −1, a))
. Now let λ be an arbitrary element of Z[H ×A], and recall that ˜C(λ1 +λ2) = ˜C(λ1) ˜C(λ2), so that ˜C(FrA(λ)) = ˜C
P
(h,a)∈H×Aλh,aFrA(h, a)
=Q
(h,a)∈H×A
C˜(FrA(h, a)) λh,a
. Then use our result for singleton sets to obtain ˜C(FrA(λ)) = Q
(h,a)∈H×AFr
C˜((h, a))λh,a
= Fr
Q
(h,a)∈H×A
C˜((h, a))λh,a
= Fr C(λ)˜
.
If I is a finite set, we extend the Frobenius action FrA to Z[I ×H ×A] so that if λ∈Z[I×H×A], then (FrA(λ))i = FrA(λi). We can easily deduce what we need to know about this action on Z[I×H×A] from the above lemma.
Corollary 2.32. FrA is a permutation of Z[I ×H ×A] with FrAee0 the identity. FrA
preserves size of accounts. Furthermore, if λ ∈ Z[I ×H×A], then |(FrA(λ))i|= |λi| for all i∈I. FrA takes multisets to multisets and (FrA(λ))! =λ! for all λ∈N[I×H×A]. If {Si}i∈I is a family of q-closed sets, and if λ∈Z[I ×H×A] with λi ∈Z[H×Si] for each i, then (FrA(λ))i ∈ Z[H×Si] for each i∈I. Additionally, FrA(λ) is all-unity if and only if λ is all-unity. Further, Π FrA(λ) = (Πλ)p, so that FrA(λ) is unity-product if and only if λ is unity-product. For each i ∈ I, Σ prH([FrA(λ)]i) = pΣ prH(λi). Thus (FrA(λ))i is Delsarte-McEliece for all i∈I if and only if λi is Delsarte-McEliece for all i∈I. Suppose ci ∈ GR(pd, e)[A] for each i ∈ I. Suppose that Ci is the element of Zp[ζq0−1][A] with C˜i=τ◦c˜i for eachi∈I. Ifλ∈Z[I×H×A], thenQ
i∈IC˜i([FrA(λ)]i) = Fr Q
i∈IC(λ˜ i)
.