The first is the study of the entropy region: the space of all possible vectors of joint entropies that can arise from a set of discrete random variables. As a result, many popular Reed-Somom decoders – the Berlekamp-Massey decoder, for example – use polynomial arithmetic to correct errors in codewords.
Entropy Vectors and the Ingleton Inequality
1.1) Although it is known that there are entropy vectors in Γ∗4 that violate Ingleton's inequality, it is not immediately clear how to construct the random variables that produce them, or how and when they can be incorporated into network codes. In Chapter 2, we show how to construct entropy vectors that violate the Ingleton group using subgroups of projective linear groupsP GL(n, q) and general linear groupsGL(n, q) for certain values of nandq.
Low-Coherence Frames
Since these are matrix groups, we are able to give concrete characterizations of the elements in each of the subgroupsGi, i= 1, ..,4. The motivation behind this is that it is known that frames of the form (1.2) achieve the lowest possible coherence for given dimensions and when all the mutual inner products between the frame elements have the same size.
Constrained Coding
Thus, if one of the systematic symbols is lost, it can only be recovered by accessing the code symbols that protect its local group (along with the remaining systematic symbols in its group). We will derive bounds for the minimum distance bounded codes, reminiscent of the cut set-type bounds of [48], and we will improve these bounds in case we need a systematic code.
Group-Characterizable Entropy Vectors
Its entropy is therefore the same as that of the random variable ΛGα, so we will identify it with the random variable Xα. Interestingly, it appears that any entropy vector can be approximated by a scaled group characteristic entropy vector [23].
Matroidal Bounds on the Entropy Region
Although this process can indeed approximate any entropy vector, and presumably can be used to enable us to study the entropy region using group-theoretic techniques, it often requires the setT (and consequently the permutation groupG) to cluster very large. However, they do not define the entropy region for all n, and in fact, as we will discuss shortly, they do not even completely describe the closure of the cone of representative matroidal rank vectors.
The Ingleton Inequality
The fact that all abelian groups must satisfy Ingleton's inequality can be understood as a generalization of the fact that linear subspaces of a vector space must satisfy it in its original form (2.7). In light of this, we want to identify examples of small non-Nebelian groups that produce entropy vectors that violate Ingleton's inequality.
Group Network Codes
For linear codes, since the superordinate groupGis abelian, we know from Theorem 2 that the subvector associated with any four of these random variables must satisfy the Ingleton inequality. Thus, by characterizing Ingleton-violating groups, we could potentially develop group network codes that are more powerful than linear codes in the sense that their associated random variables can achieve a larger area of the entropy region.
The Smallest Ingleton-Violating Groups: P GL(2, p)
Without too much difficulty, the Ingleton violation from the previous section can be generalized [70] to produce a violating set of subgroups in any projective linear group PGL(2, q) forq a prime power greater than or equal to 5. In other terms of the Ingleton inequality, the intersections G12, G13 and G14 take the same forms as in (2.23), and all have size 2.
Ingleton Violations in GL(2, q)
- Instance 1: The Preimage Subgroups
- Variants of the Preimage Subgroups with Different G 1
- Variants of the Preimage Subgroups with Different G 2
- The Final Four Ingleton Violations
As described earlier, each of these is a semidirect product of a normal subgroup of GA with a cyclic group. We have described the shape of the elements in each of the subgroups G1, G2 and G3 and.
Interpreting the Ingleton Violations Using Group Actions
Ingleton Violations in More General 2-Transitive Groups
We will talk about the coherence of M to be the coherence of the frame {mi}. This version of the theorem is typically proved (e.g. in [91]) by considering the eigenvalues of the Gram matrix G := [hfi, fji].
Frames from Unitary Group Representations: Slepian Group Codes
In what follows, we discuss constructions of unit-rate tight frames in which we can control the number of distinct values of the inner product and ensure that each arises with the same multiplicity. Thus, in practice, depending on our set U, the Slepian construction can actually give us up to n−12 distinct non-trivial values of the inner product, although it is important to note that these may not arise with the same plurality when grouped together in this fashion.
Abelian Groups and Harmonic Frames
Remark: Note that the inner products corresponding to Uk and U−1k actually have the same rates, since. It is not too difficult to see that the frame matrix M is a row subset of the KroneckerAKron product.
Equiangular Frames from Cyclic Group Representations
This concept of tight equiangular frames arising from difference sets has since been generalized and elaborated. Many of our results in the following sections can also be seen as relaxing difference sets even further to produce low coherence frames.
Cyclic Groups of Prime Order
So, the total number of distinct pairwise inner products we now need to control is n−1m. It remains only to show that each of the n−1m inner products occurs the same number of times.
Sharper Bounds on Coherence for Frames from Cyclic Groups of Prime Order
In (b), as expected, there are only three different values of the inner products between different, normalized columns. In Appendix B.1 we provide an alternative classification for when exactly this last coherence limit applies.
Optimizing Coherence Over Cosets
However, there is value in searching over these cosets, as this search will converge to its optimal value much faster than the search over cosets in the smaller group. Of course, we can still bind the context of the frames that can arise from this construction, using our previous frames.
Generalized Dihedral Groups
Simulating Generalized Dihedral Frames with Harmonic Frames
It turns out that in this case we could have created frames with the same dimensions whose inner products have exactly the same order of magnitude as those of the above generalized dihedral frames if we had replaced Tand [τ] with . Together with [σ], this is no longer a representation of a generalized dihedral group, but rather a representation of the abelian group nZZ× DZZ.
Summary
We define the group Fourier transform of a complex-valued function on G,f:G→C as the function that maps a graded representationρ onto the d×dcomplex matrix. LetG={gi}ni=1 be a finite group with inequivalent, irreducible representations{ρi}ni=1r , andF be the group Fourier matrix of Gas in (4.4).
Reducing the Number of Distinct Inner Products in Tight Group Frames
Analogous to equations (4.7) and (4.8) from the proof of Theorem 16, this corresponds to the tight group frame whose elements are the images of the vector v. We have thus formed M by choosing the rows of the Fourier group. matrix corresponding to a subset of representations of the form{σi·ρ}, where the{σi}form a subset of automorphisms.
Choosing the Automorphism Subgroup
Then for any σ1, σ2∈H that are in the same right bank set of KA, the inner products associated with σ1(g) andσ2(g) are respectively equal for any g∈G. Since σ1andσ2 are in the same right composition of KA (which is equal to AK), there is something∈H such thatσ1=a1k1handσ2=a2k2hfor somea1, a2∈A and somek1, k2∈K.
Subgroups and Quotients of General Linear Groups
Frames from Vector Spaces Over Finite Fields
We must point out that the general form of an irreducible representation of the additive group of Fpr is ρa(x) := ωTr(ax), where ω = e2πip and a ∈ Fpr. To correct this problem, we will shift our focus to the interpretation of Gas, the additive group in the field Fpr.
Smaller Alphabets and Frames from Hadamard Matrices
Difference Sets
The columns of Min (4.44) form a rigid equiangular frame if and only if the elements in A = {ai}mi=1 form a difference set in Fpr. Then the columns of M in (4.44) form a rigid equiangular frame whose entries are each one of the separate roots of unity.
Frames from Special Linear Groups
Frames from Induced and Cuspidal Representations
The following theorem uses our previous results on frames constructed from finite fields to give a bound on the coherence of the frames we can construct from the induced representations of SL2(Fq), for q even. To complete the proof, we simply need to take the maximum of the inner product sizes corresponding to the elements u and w`.
Satisfying the Strong Coherence Property
Although the frame matrixF can be written out concretely using the explicit forms of the representations given in , we will omit this process because we have already described it in depth and because these particular frames tend to have fairly large dimensions. Equations (4.12) and (4.13) show that after normalization of the columns of M, the inner product between the ith and the jth column is the same.
Summary
Prior Work
The problem of constructing error-correcting codes with constrained coding has been addressed by various authors. The problem concerns a group of users, each with a subset of messages, who want to broadcast their information securely when an eavesdropper is present.
Problem Setup
The choice of support entries for a valid generator matrix G determines the order of the code (which can be between 0 and s) and its minimum distance. Additionally, we would like to provide efficient methods to decrypt our code words in the presence of errors.
Minimum Distance Bounds for General and Constrained Codes
This proof is essentially a variation of the proof of the Singleton bound when restricted to the code induced by the subvectors [c]N(I),c∈ C, consisting of the codewords in C with their coordinates removed outside the setN(I) . Note: In the case where |N(I)| < |I| for a subset I, then the proof above can produce two distinct vectors sm1 and m2 in SI, giving c1 and c2 in CI, which have the same entries in all coordinates of N(I), and thus c1=c2, so our code has minimum distance equal to 0.
Subcodes of Reed-Solomon Codes
The dimension of codeCis is equal to the rank of T, which is determined by how we choose the polynomial path(x) subject to the aforementioned requirements that deg(ti(x))< kandQ. Furthermore, since C is a subspace of a Reed-Solomon code, we can use already existing efficient decoders to recover any messagem∈Fsq from the codeword c:=mG, even in the presence of errors.
Systematic Codes
Systematic Code Construction Using Reed-Solomon Codes
LetAE˜be the adjacency matrix of the graph ˜G:= (M,V,E \Eneg), which is the graphG after removing the ignored edges. Note that any code that satisfies the constraints imposed by ˜G will automatically fit that of the original graph G.
Minimum Distance for Systematic Linear Codes
The eGrows generate a code with minimum distance at least that of the original Reed-Solomon code, which is n−k+ 1. Moreover, by setting k=ksys for our Reed-Solomon code, we see that this new code C has distance minimal at least n−ksys+ 1 = dsys.
Arbitrary MDS Codes
Another thing to point out is that in the case where we want C to be a system code such that the columns of its generator matrixG are a permutation of the columns of the xidentity matrix, we have to be a bit more careful in choosing hi .
Example
If we specify that x1 is any of the m elements of A, we have exactly NA,A choices for x2 that satisfy this condition (for each a∈A such that 1 +a∈A, simply set x2=ax1ζ−1) . Let G=Fpr be a finite field with elements {x1, .., xpr} and let H =F×pr be a (cyclic) non-zero multiplicative group of elements of the field.
