The Gleason--Prange theorem

In total, the computation of the Fourier transform requires a total of fifty-two multipli-cations (3×4²+2²) in the field GF(16). Some of these multiplications are by 1 and can be skipped. In contrast, direct computation of the Fourier transform as defined requires a total of 225 multiplications in the field GF(16). Again, some of these multiplications are by 1 and can be skipped.

1.11 The Gleason--Prange theorem

The final two sections of this chapter present several theorems describing proper-ties, occasionally useful, that are unique to Fourier transforms of prime blocklength.

Although these properties are only of secondary interest, we include them to satisfy our goal of presenting a broad compendium of properties of the Fourier transform.

Let p be an odd prime and let F be any field that contains an elementω of order p, or that has an extension field that contains an elementω of order p. This requirement is equivalent to the requirement that the characteristic of F is not p. Let v be a vector of blocklength p. The Fourier transform of blocklength p,

V_j =

p−1

i=0

ω^ijvi j= 0, . . . , p − 1,

43 1.11 The Gleason--Prange theorem

has, of course, all the properties that hold in general for a Fourier transform. Moreover, because the blocklength is a prime integer, it has several additional properties worth mentioning. These are the Gleason–Prange theorem, which is discussed in this section, and the Rader algorithm, which is discussed in Section1.12.

The indices of v and of V may be regarded as elements of GF( p), and so we call GF( p) the index field. The index field, which cannot contain an element of order p, should not be confused with the symbol field F. The elements of GF( p) can be partitioned as

GF( p) =Q∪N ∪ {0},

whereQis the set of (nonzero) squares (called the quadratic residues) andN ^{is the} set of (nonzero) nonsquares (called the quadratic nonresidues). Not every element of GF( p) can be a square because β²= (−β)². This means that two elements of GF( p) map into each square. Not more than two elements can map into each square because the polynomial x²− β² has only two zeros. Thus there must be( p − 1)/2 squares.

This means that there are( p − 1)/2 elements inQ^and( p − 1)/2 elements inN^{. If} π is a primitive element of GF( p), then the squares are the even powers of π and the nonsquares are the odd powers ofπ. This partitioning of the index set into squares and nonsquares leads to the special properties of the Fourier transform of blocklength p.

The Gleason–Prange theorem holds in any field, but the statement of the general case requires the introduction of Legendre symbols and gaussian sums, which we prefer to postpone briefly. Initially, to simplify the proof, we temporarily restrict the treatment to symbol fields F of the form GF(2^m).

The Gleason–Prange theorem deals with a vector v of blocklength p, with p a prime, augmented by one additional component, denotedv_∞. With this additional component, the vector v has length p+1. For the field GF(2^m), the additional component is given by

v_∞=

p−1

i=0

vi = V0.

The Gleason–Prange permutation of the vector v= (v0,v1,v2,. . . ,vp−1,v∞)

is the vector u with the components u_i=v_−i⁻¹, and with u₀=v_∞^{and u}_∞=v0. The index−i⁻¹is defined in terms of the operations of the field GF( p). If the Gleason–

Prange permutation is applied twice, the original v is restored because−(−i⁻¹)⁻¹ = i in GF( p).

For example, with p= 11, the Gleason–Prange permutation of the vector v= (v0,v1,v2,v3,v4,v5,v6,v7,v8,v9,v10,v∞)

is the vector

u= (v∞,v10,v5,v7,v8,v2,v9,v3,v4,v6,v1,v0).

The Gleason–Prange permutation of the vector u is the vector v.

We shall say that the spectrum V satisfies a Gleason–Prange condition if either V_j = 0 for every j ∈ Q^{, or V}j = 0 for every j ∈ N. For example, for p = 11, Q= {1, 4, 9, 5, 3} andN = {2, 6, 7, 8, 10}, so both the vector

V = (V0, 0, V₂, 0, 0, 0, V₆, V₇, V₈, 0, V₁₀) and the vector

V = (V0, V₁, 0, V₃, V₄, V₅, 0, 0, 0, V₉, 0) satisfy a Gleason–Prange condition.

Theorem 1.11.1 (Gleason–Prange) Over GF(2^m), suppose that the extended vectors v and u are related by the Gleason–Prange permutation. If V satisfies a Gleason–

Prange condition, then U satisfies the same Gleason–Prange condition.

Proof: We shall prove the theorem for the case in which Vj = 0 for j ∈Q. The other case, in which V_j= 0 for j ∈N, is treated the same way.

Because V₀=v∞, the inverse Fourier transform of V can be written as follows:

vi=v_∞+

p−1

k=1

ω^−ikV_k.

Consequently,

v_−i⁻¹=v∞+

p−1

k=1

ω^i−1kV_k i= 1, . . . , p − 1.

On the other hand, because u₀ = v∞, the Fourier transform of u can be written as follows:

U_j = u0+

p−1

i=1

ω^iju_i j= 1, . . . , p − 1,

=v∞+

p−1

i=1

ω^ijv_−i^−1,

45 1.11 The Gleason--Prange theorem

Combining these equations, we obtain

U_j =v_∞ equal, so everyω^roccurs twice in the formula for U_j, and the proof is complete.

The theorem holds because the array A_jk = 

iω^ij+i−1k has an appropriate pattern of zeros. To illustrate an example of this pattern, let p= 7, and let ω be an element of GF(7) that satisfies ω³+ ω + 1 = 0. Then it is straightforward to calculate the array

By the permutation of its rows and columns, this matrix can be put into other attractive forms. For example, the matrix A can be put into the form of a block diagonal matrix with identical three by three matrices on the diagonal and zeros elsewhere. Alternatively, the matrix can be written as follows:

⎡

with each row the cyclic shift of the previous row. Then the Gleason–Prange theorem, stated in the Fourier transform domain, becomes obvious from the above matrix–

vector product. A similar arrangement holds for arbitrary p, which will be explained in Section1.12as a consequence of the Rader algorithm.

The Gleason–Prange theorem holds more generally for a Fourier transform of block-length p in any field F whose characteristic is not equal to p, provided the definition of the Gleason–Prange permutation is appropriately generalized. For this purpose, letθ denote the gaussian sum, which in the field F is defined for anyω of prime order p by

θ =

p−1

i=0

χ(i)ωⁱ,

whereχ(i) is the Legendre symbol, defined by

χ(i) =

⎧⎪

⎨

⎪⎩

0 if i is a multiple of p

1 if i is a nonzero square(mod p)

−1 if i is a nonzero nonsquare (mod p).

An important property of the Legendre symbol for p prime that we shall use is that

p−1

i=0

χ(i)ω^ij= χ( j)θ,

which is easy to prove by a change of variables using GCD( j, p) = 1.

Theorem 1.11.2 For any field F whose characteristic is not p, the gaussian sum