These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight multipliers of linear codes over rings. This means that C can be obtained as a linear combination of the sequences of Control the Galois ringGR(p`, m).
History
The gray mapping above Zp` is a distance-preserving mapping from Znp` to Fnpp `−1, where we take the homogeneous weight in Znp` and the Hamming weight in Fnpp `−1. In 1973, in his work [5], Delsarte considered the problem of MacWilliams identities for Hamming weighted enumerators of abelian groups.
Methods and Summary of the Main Results
This turns out to be a generic formula for the MacWilliams identities for the complete weights of the group codes. A similar result was obtained for the symmetric weight adders of linear codes over Galois rings inspired by the homogeneous weight.
Galois Rings
Linear Codes over Galois Rings and the Homogeneous Weight
So, in this sense, we can consider the homogeneous weight as an extension of the Lee weight on. Several authors have used this relation of the homogeneous weights with the exponential sums to obtain results on linear codes over Galois rings.
The Main Results
For the remainder of this section, we will let C be a linear code of the same type as in Theorem 2.11, and ˜C be the same code as defined in the proof above. Suppose that C has the same generators as in the proof of Theorem 2.11, and let ˜C be the same code as defined in the proof of Theorem.
The Result in the Main Theorem is Best Possible
What Theorem 2.16 accomplishes is that it proves that the result in Theorem 2.11 is best possible in all cases except possibly the case where e=`, p= 2, m= 1. The results in Theorem 2.11 and Theorem 2.13 that we obtained in section 2.4 are best possible in all cases.
Concluding Remarks and Questions
Using the MacWilliams identities for these weight counters, they were able to establish the MacWilliams identities for the weight counters of Hamming and Lee linear codes over Z4. In Section 2, we will consider codes of general abelian groups and prove the MacWilliams identity theorem for complete weight numerators of these codes. 3.9) Now our goal is to determine whether there is a similar MacWilliams identity as those above for Euclidean weight numerators of linear codes over Z4.
Since the MacWilliams identities exist for the Hamming and Lee weight counters of linear codes over Z4 of the form (3.8) and (3.9), we naturally ask the same question for the Euclidean weight counters of linear codes over Z4. Note that Theorem 3.2 implies that we cannot have a MacWilliams-like identity for the Euclidean weight counters of linear codes over Z4, unlike the Hamming and Lee weight counters for which we have the identities (3.8) and (3.9).
MacWilliams Identities for Group Codes
Taking all but one ofgi's to be the zero element in G, we can reduce it to the . As we said earlier, knowing the complete weight lifter makes it easier to calculate all the other weight lifters. Now, as an application, we will obtain the MacWilliams identity for the Hamming weight adders of Abelian group codes using Theorem 3.10, which will be exactly the same result as Delsarte obtained in [5].
In fact, we note that it can be written in terms of the complete weight enumerator as. Then we have the Macwilliams identity for the Hamming Weight enumerators of C and C∗ as follows:
MacWilliams Identities for Linear Codes over Rings
Then we define the dual C∗ of C with respect to the inner product defined in (3.34) as. ª is the Galois Ring expansion of Zp`, and suppose that C is a linear code over GR(p`, m) of length n and let C∗ be the dual of C with respect to the inner product defined in (3.34) ). Suppose that C∗ is the dual of C with respect to the inner product defined in (3.34), and let γ be a primitive depth root of unity.
We define the inner product on this ring, so that the dual of linear codes will also be linear. Then, by calculating the corresponding ai∗ajs in Corollary 3.24, we see that if C∗ is dual C with respect to the inner product defined in (3.43) and (3.44), then we obtain.
Concluding Remarks and Some Applications
As an example, we'll go back to the code that gave us the counterexample in Section 3.1. In section 1, we will use a Gray map from Z9 to Z33 and similar techniques to those used in [2] and [8] to obtain some nonlinear ternary codes with comparably high minimum distances. In section 3, we will give a purely combinatorial construction of the Gray map we defined earlier, using the affine geometries.
In Section 4, we will talk about the Gray map for the ring F2m+uF2m and obtain some results about the Lee weights of linear codes on these rings. We will use the same technique that Carlet used in [8] to obtain some nonlinear ternary codes as images.
This shows that our construction could lead to ternary codes with relatively large minimum distances, especially if the code is large. The main result of this section is to prove the distance-preserving property of this map. The proof consists of exhausting all cases and verifying the distance-preserving property for all of them.
Now using the definition of Gk and the hypothesis that Gk−1 conserves distance, we see that p−1, which by induction hypothesis means that. This means that by the induction hypothesis we have modp), which by induction hypothesis means that.
A Combinatorial Construction of the Gray Map
The result of Lemma 4.13 implies that hyperplanes containing none of the lines L0, L1,. Lpk−1−1, or equivalently there are exactly hyperplanes pk containing none of the linesL0, L1,. Since, from Lemma 4.13, we know that hyperplanes containing none of the lines L0, L1,.
This means that the number of parallel classes of hyperplanes that do not contain any of the lines L0, L1,. Γpk−1−1 are parallel classes of hyperplanes that do not contain any of the lines L0, L1,.
The PeshawBL of x=a+ub∈Rm is defined to be the sum of the Lee weight of b∈F2m and that of a+b∈F2m. This proves the first part of the lemma, and the second part of the lemma just follows from the definition of Lee's weight in Rm. So the Lee weight with respect to B of the shape coordinate set.
It is easy to see that any codeword in Ck2 has a Lee weight divisible by 4, which means that modulo z4−1 we will have. Although F2+uF2 is very similar to Z4, the extension of the Lee weight to the ring F2m+uF2m is not a homogeneous weight, unlike the extension of the Lee weight from Z4 to Galois rings that we used in Chapter 2.
Permutation Invariance and Linearity over Rings
Suppose that C is a binary linear code of length 2n for some n and let C⊥ be its dual. Suppose that a binary linear codeC of length 2n is the image under the Gray map of a linear codeDoverF2+uF2 of lengthn. As usual, we define K4-invariance of a binary code C of length 4n as invariant under the permutations α, β, γ.
Note that ϕ is a linear map and it maps a linear code over F2 +uF2 +vF2 +uvF2 of length n to a linear binary code of length 4n. For the second implication, suppose we have a binary linear code of length 4n that is K4-invariant.
Reed-Muller Codes and Permutation Invariance
According to the calculations above, C being invariant with respect to β implies that D is invariant with respect to left multiplication by 1 +u and hence u. Finally, the invariance of C under γ implies that it is invariant under left multiplication by (1 +u+v+uv), but since it is already invariant under multiplication by uinvan and because it is additive, it implies that it is invariant under multiplication by uv. The only remaining example is to look at a vector of the form vm−1∗vm∗vi1∗ · · · ∗vis, but then from the above discussion we see that.
Since ρ takes all generators of RM(r, m) into vectors inRM(r, m), we can conclude by Lemma 5.4 that RM(r, m) is invariant under the permutation ρ or. By the same reasoning as above in the case of Z4, we can easily see that all the generators of RM(r, m) with 0≤r≤mare mapped to vectors inRM(r, m), proving that RM(r, m ) is invariant under the action of the permutation group K4 of Lemma 5.4.
The Pre-images of Reed-Muller Codes under Gray Maps
As we see, this turns out to be another major difference between Z4 and F2 +uF2, since another conclusion can be stated as follows which contrasts with Theorem 8 in [2] for the case of Z4. We are curious to find these linear codes that give us the Reed-Muller codes in this section. In the same way as we did above, we can find the pre-images of the Reed-Muller code as linear codes over the ringsF2+uF2+u2F2+u3F2 and F2+uF2+vF2+uvF2.
We will summarize them in the following theorems, but omit the proofs because they are very similar to the proof of Theorem 5.26. Let DRM(r, m) be the linear code over F2+uF2+vF2+uvF2 of length 2m−2 generated by the codes.
Reed-Muller Codes and the Permutation Group Z 2 k
To obtain a result about the Z2k invariance of RM(r, m), we will need to impose some restrictions on k, r, and m. To do this we will first start with the basic generators of RM(r, m) and use techniques similar to those we used in the previous section. We will need to prove that applied to any generator of RM(r, m) will still be inRM(r, m).
Now suppose that m−k≤r−2. Then we will take=m−k, which means that, in this case, we will have. This means that the discussion on the permutation invariance of Reed-Muller codes under the permutations in Section 5.3 is in some sense complete.
Then C is invariant with respect to the permutation group Z2k if and only if C is the image under φk of some linear code over Sk of length n. This means that C is invariant with respect to τ, which means that it is invariant with respect to τi for alli= 0.1,. Conversely, suppose that C is a binary linear code of length 2kn that is invariant under the operation of the permutation group Z2k.
For this, all we need to show is that D is invariant under multiplication from the left by powers ofu. But then, the fact that D is additive and an easy induction leads us to conclude that Dis invariant under multiplication from the left by ui, i= 0,1,.
