DISEÑO DE METODOLOGÍA PARA AUDITORÍA DEL PLAN DE ENTRENAMIENTO

A cyclic code over Z₄can be studied entirely in terms of the polynomials of the ring Z₄[x]. However, just as it is productive to study codes over the field GF(q) in the larger algebraic field GF(q^m), so, too, it is productive to study codes over Z4 in a larger algebraic system called a Galois ring. A Galois ring over Z₄ is defined in a way analogous to the definition of an extension field of GF(q). Let h(x) be a primitive basic irreducible polynomial (a primitive polynomial) of degree m over Z₄. Then, with the natural definitions of addition and multiplication, Z₄[x] is a ring of polynomials over Z₄, and the Galois ring Z₄[x]/
h(x) is the ring of polynomials modulo h(x).

This Galois ring has 4^melements, and is denoted GR(4^m). Although some properties of

Table 2.6. The cycle of a primitive element in GR(4^m)

ξ¹= x ξ²= x²

ξ³= 2x²− x + 1 ξ⁴= −x²− x + 2 ξ⁵= x²− x − 1 ξ⁶= x²+ 2x + 1 ξ⁷= 1 = ξ⁰.

Galois fields carry over to Galois rings, other properties do not. In particular, the Galois ring GR(4^m) cannot be generated by a single element. However, there will always be an element with order 2^m, which we will callξ. It is a zero of a primitive polynomial over Z₄, and hence may be called a primitive element, though it does not generate the Galois ring in the manner of a primitive element of a Galois field. Ifξ is a primitive element of GR(4^m), then every element of GR(4^m) can be written as a + 2b, where a and b are elements of the set{0, 1, ξ, ξ²,. . . , ξ^2m−2}. Because 2^m· 2^m = 4^m, this representation accounts for all 4^melements of the ring GR(4^m). With the convention thatξ^−∞ = 0, every element of GR(4^m) can be written in the biadic representation as ξⁱ+ 2ξ^j.

For example, to construct the Galois ring GR(4³), choose the primitive polynomial x³+ 2x² + x − 1 over Z4. Then let ξ = x, and write the cycle of ξ, as shown in Table2.6. The 64 elements of GR(64), then, are those of the form a + 2b, where a, b ∈ {0, 1, ξ, ξ²,. . . , ξ⁶}. Of course, the biadic representation is not the only representation.

Each element of GR(64) can also be written as a polynomial over Z4in x of degree at most 6, with multiplication modulo h(x).

It is now an easy calculation in this Galois ring to verify the following factorizations:

x³+ 2x²+ x − 1 = (x − ξ)(x − ξ²)(x − ξ⁴), x³− x²+ 2x − 1 = (x − ξ³)(x − ξ⁶)(x − ξ⁵),

x− 1 = (x − ξ⁰).

Each such factorization can be regarded as a kind of lift to GR(4³) of a like factorization over GF(2³). The primitive element ξ of GR(4^m) becomes the primitive element α of GF(2^m) when GR(4^m) is mapped into GF(2^m). This means that the cyclic orbit of ξ, taken modulo 2, becomes the cyclic orbit ofα.

In general, the elements of the Galois ring GR(4^m) may be represented in a variety of ways. One, of course, is the definition as

iaixⁱ, a polynomial in x of degree at most

115 2.15 Galois rings

Table 2.7. Galois orbits in GR(4^m) and GF(2^m)

ξ¹= x α¹= x

ξ²= x² α²= x²

ξ³= x + 1 +2(x²+ x) α³= x + 1 ξ⁴= x²+ x +2(x²+ x + 1) α⁴= x²+ x ξ⁵= x²+ x + 1 +2(x + 1) α⁵= x²+ x + 1 ξ⁶= x²+ 1 +2x α⁶= x²+ 1

ξ⁷= 1 = ξ⁰ α⁷= 1 = α⁰

m− 1. We have also seen that we may write an arbitrary ring element, β, in the biadic representation

β = ξⁱ+ 2ξ^j

= a + 2b,

where a and b, or a(β) and b(β), denote the left part and right part of β, respectively.

Each part is a power ofξ. This representation is convenient for some calculations. As a third representation, it may be helpful to see the elements of GR(4^m) lying above the elements of GF(2^m). For this purpose, regard the element β of GR(4^m) to be written asβo+ 2βe, whereβoandβe, called the odd part and the even part of the ring element β, are both polynomials in x with all coefficients from {0, 1}.

To find the representationβ = βo+ 2βe, write the odd part asβo = β modulo 2, then the even partβeis determined as the difference betweenβ and βo. With due care, bothβoandβecan be informally regarded as elements of the extension field GF(2^m), though operations in GR(4^m) are actually modulo 4, not modulo 2.

To see the relationship betweenξ and α, the comparison of the cycles of ξ and α, given in Table2.7, is useful: The cycle ofξ is the same as in Table2.6, but expressed to show the role of the two. We may summarize this relationship by writingξ^j = α^j+2γj, where 2γj is defined asξ^j− α^j. Thus α^j is the odd part ofξ^j andγj is the even part ofξ^j.

The following proposition tells how to calculate the representationξⁱ+2ξ^jfrom any other representation ofβ.

Proposition 2.15.1 Letβ = a + 2b denote the biadic representation of β ∈ GR(4^m).

Then a= β^2m and

2b= β − a.

Proof: To prove the first expression, observe that β²= (a + 2b)²

= a²+ 4ab + 4b²(mod 4)

= a².

Because a is a power ofξ, and so has order dividing 2^m− 1, repeated squaring now givesβ^2m = a^2m = a, which is the first expression of the proposition. The second

expression is then immediate.

Proposition 2.15.2 Let a(β)+2b(β) be the biadic representation of any β ∈ GR(4^m).

Then

a(β + γ ) = a(β) + a(γ ) + 2(βγ )^2m−1, a(βγ ) = a(β)a(γ ).

Proof: Using Proposition2.15.1, the statement to be proved can be restated as (β + γ )^2m = β^2m+ γ^2m+ 2(βγ )^2m−1.

For m= 1, this is elementary:

(β + γ )²= β²+ γ²+ 2βγ .

Because 4= 0 in this ring, it is now clear that

(β + γ )⁴= (β²+ γ²)²+ 4(βγ )(β²+ γ²) + 4(β²γ²)

= (β²+ γ²)²

= β⁴+ γ⁴+ 2β²γ².

The recursion is now clear, so the proof of the first identity is complete. The proof of the second identity follows fromβγ = (a + 2b)(a^+ 2b^) = aa^+ 2(ab^+ a^b).
The statement of Proposition2.15.2will now be extended to the generalization in which there are n terms in the sum.

Proposition 2.15.3 Let a(β
) + 2b(β
) denote the biadic representation of β
∈ GR(4^m). Then

a


n

=1

β



=

n

=1

a(β
) + 2

n

=1

^=

(β
β
^)^2m−1.

117 2.15 Galois rings

Proof: If there are two terms in the sum, the statement is true by Proposition2.15.2.

Suppose that the expression is true if there are n− 1 terms in the sum. Then

a of b because 4 = 0 in this ring. In this sense, squaring is a lossy operation. A useful variant of the squaring function is the frobenius function, defined in the Galois ring GR(4^m) as c^f = a² + 2b². Now the trace in GR(4^m) can be defined as tr(c) = c+ c^f+ · · · + c^fm−1.

There is also a Fourier transform in the Galois ring GR(4^m). A “vector” c of blocklength n= 2^m− 1 over the ring GR(4^m) has a Fourier transform, defined as

C_j= Fourier transform C is also a vector of blocklength n over the ring GR(4^m). Because Z₄ is contained in GR(4^m), a vector c over Z4 of blocklength n is mapped into a vector C over GR(4^m) by the Fourier transform. Moreover, by setting 2 = 0, the Fourier transform in the ring GR(4^m) can be dropped to a Fourier transform in the field GF(2^m), with components Cj =_n−1

i=0 α^ijc_i.

Many elementary properties of the Fourier transform hold for the Galois ring Fourier transform. The inverse Fourier transform can be verified in the usual way by using the relationship

unless ξ = 1. Therefore, because an inverse Fourier transform exists, each c corresponds to a unique spectrum C.

There is even a kind of conjugacy relationship in the transform domain. Let c be a vector over Z₄, written c = a + 2b, with components displayed in the biadic rep-resentation as c_i = ai+ 2bi. Because c is a vector over Z₄, the components satisfy

Although C_jis not itself in the biadic representation, each term within the sum is in the biadic representation, because a_iand b_ican only be zero or one.

We now express the spectral component C_jin the biadic representation as C_j = Aj+ 2B_j. By Proposition2.15.3, the left term of the biadic representation of C_j =

ic_iξ^ij

Because 4= 0 in this ring, the second term can be simplified so that A_j =

is the biadic representation of C_j.

Although this representation for C_j seems rather complicated, it is the starting point for proving the following useful theorem. This theorem characterizes the spectral com-ponents of a vector over Z₄. In particular, the theorem says that component C_2j, which is given by

119 2.15 Galois rings

is related to C_jby a conjugacy constraint. The theorem also implies that if C_j = 0, then C_2j= 0 as well.

Theorem 2.15.4 Let c be a vector of blocklength n= 2^m− 1 over Z4. Then the com-ponents of the Fourier transform C satisfy C_2j = C_j^f, where C_j^f denotes the frobenius function of C_j.

Proof: We will give an explicit computation using the formula derived prior to the statement of the theorem. Write

C_j^f =

Now rewrite each of these three squares. The first square is expanded as



Each of the second two squares can be expanded in this way as well, but the cross terms drop out because 4 = 0 in the ring Z4. The summands in these latter two terms then become(biξ^ij)²and((aia_iξ^ijξ^ij)^2m−1)². Therefore because each a_ior b_ican only be a

This theorem allows us to conclude that, as in the case of a Galois field, if g(x) is a polynomial over Z4with a zero at the elementξⁱof the Galois ring GR(4^m), then it

also has a zero at the elementξ²ⁱ. In particular, a basic irreducible polynomial over Z₄, with a zero atβ, has the form

p(x) = (x − β)(x − β²) . . . (x − β^2r−1),

where r is the number of conjugates ofβ in GR(4^m).

A cyclic code over GR(4^m) that is defined in terms of the single generator polynomial g(x) consists of all polynomial multiples of g(x) of degree at most n − 1. Every codeword has the form c(x) = a(x)g(x).Although g(x) is the Hensel lift of a polynomial over GF(2^m), a(x)g(x) need not be the Hensel lift of a polynomial over GF(2^m). In particular, not every a(x) over GR(4^m) is the Hensel lift of a polynomial over GF(2^m).

One way to define a cyclic code over Z₄– but not every cyclic code over Z₄– is as the set of polynomials in Z₄[x]/
xⁿ−1 with zeros at certain fixed elements of GR(4^m).

This is similar to the theory of cyclic codes over a field. For example, the cyclic code with the primitive polynomial x³+ 2x²+ x − 1 as the generator polynomial g(x) can be defined alternatively as the set of polynomials over Z₄of degree at most 7 with a zero at the primitive elementξ, a zero of g(x). Thus c(x) is a codeword polynomial if c(ξ) = 0. Then Theorem2.15.4tells us that c(ξ²) = 0 as well, and so forth.

In the case of a cyclic code over a Galois field, the generator polynomial g(x) can be specified by its spectral zeros. Similarly, a single generator polynomial for a cyclic code over Z₄ can be specified by its spectral zeros. Because the spectral zeros define a simple cyclic code over Z₄, the minimum distance of that code is somehow implicit in the specification of the spectral zeros of the single generator polynomial. Thus, we might hope for a direct statement of this relationship analogous to the BCH bound.

However, a statement with the simplicity of the BCH bound for a Lee-distance code over Z₄is not known. For this reason, it is cumbersome to find the minimum distance of a cyclic code over Z₄that is defined in this way.

A cyclic code over Z₄can be dropped to the underlying code over GF(2), where the BCH bound does give useful, though partial, information about the given Lee-distance code. If codeword c over Z₄is dropped to codeword c over GF(2), then the codeword c will have a 1 at every component where the codeword¯c has either a 1 or a 3. Hence the minimum Lee distance of the Z₄code is at least as large as the minimum Hamming distance of that binary code, and that minimum distance satisfies the BCH bound.

Our two examples of cyclic codes over Z₄that will conclude this section are known as Calderbank–McGuire codes. These codes over Z₄are defined by reference to the Galois ring GR(4⁵). They are related to the binary (32, 16, 8) self-dual code based on the binary(31, 16, 7) cyclic BCH code, in the sense that the Calderbank–McGuire codes can be dropped to these binary codes. The cyclic versions of the two Calderbank–McGuire codes are a(31, 18.5, 11) cyclic Lee-distance code over Z4 and a(31, 16, 13) cyclic Lee-distance code over Z₄. When extended by a single check symbol, these cyclic codes over Z4are, respectively, a(32, 18.5, 12) Lee-distance code over Z4and a(32, 16, 14)

121 2.15 Galois rings

Lee-distance code over Z₄. When the symbols of Z₄ are represented by pairs of bits by using the Gray map (described in Section 2.16), these codes become nonlinear (64, 37, 12) and (64, 32, 14) binary Hamming-distance codes, with datalengths 37 and 32, respectively. Their performance is better than the best linear codes known. The comparable known linear codes are the(64, 36, 12) and (64, 30, 14) BCH codes, with the dimensions 36 and 30.

The first Calderbank–McGuire cyclic code is the set of polynomials c(x) of block-length 31 over Z₄that satisfy the conditions c(ξ) = c(ξ³) = 2c(ξ⁵) = 0, where ξ is a primitive element of GR(4⁵). The condition 2c(ξ⁵) = 0 means that c(ξ⁵) must be even, but not necessarily zero, which accounts for the unusual datalength of this(31, 18.5, 11) cyclic Calderbank–McGuire code over Z₄. Accordingly, the check matrix of this cyclic code is given by

H =

⎡

⎢⎣

1 ξ¹ ξ² · · · ξ³⁰ 1 ξ³ ξ⁶ · · · ξ⁹⁰ 2 2ξ⁵ 2ξ¹⁰ · · · 2ξ¹⁵⁰

⎤

⎥⎦ .

In the Galois ring GR(4⁵), the elements ξ, ξ³, andξ⁵each have five elements in their conjugacy classes. This means that the first two rows of H each reduce the datalength by 5. The third row only eliminates half of the words controlled by the conjugacy class ofξ⁵. Thus n− k = 12.5 and n = 31, so k = 18.5.

The cyclic(31, 18.5, 11) Calderbank–McGuire code over Z4can be lengthened by a simple check symbol to form the(32, 18.5, 12) extended Calderbank–McGuire code over Z₄. The lengthened code has the check matrix

H =

⎡

⎢⎣

1 1 ξ¹ ξ² · · · ξ³⁰ 0 1 ξ³ ξ⁶ · · · ξ⁹⁰ 0 2 2ξ⁵ 2ξ¹⁰ · · · 2ξ¹⁵⁰

⎤

⎥⎦ .

There are two noteworthy binary codes that are closely related to this code. A linear code of blocklength 32 is obtained by simply dropping the codewords into GF(2), which reduces every symbol of Z₄to one bit – a zero or a one according to whether the Lee weight of the Z₄symbol is even or odd. This map takes the Z₄code into a linear binary(32, 22, 5) code. It is an extended BCH code. The other binary code is obtained by using the Gray map to represent each symbol of Z₄by two bits. The Gray map takes the Z₄code into a nonlinear binary(64, 37, 12) code. The performance of this code is better than any known linear binary code.

If the 2 is struck from the last row of H of the cyclic code, then we have the second Calderbank–McGuire cyclic code, which has c(ξ) = c(ξ³) = c(ξ⁵) = 0 in GR(4⁵).

This gives a cyclic(31, 16, 13) Lee-distance code over Z4with datalength 16. It can be lengthened by a simple check symbol to form a(32, 16, 14) Lee-distance code over Z4.

The lengthened code has the check matrix

H =

⎡

⎢⎣

1 1 ξ¹ ξ² · · · ξ³⁰ 0 1 ξ³ ξ⁶ · · · ξ⁹⁰ 0 1 ξ⁵ ξ¹⁰ · · · ξ¹⁵⁰

⎤

⎥⎦ .

The Gray map takes the Z₄code into a nonlinear(64, 32, 14) binary Hamming-distance code.

Inspection of the check matrices makes it clear that the two cyclic Calderbank–

McGuire codes over Z₄, of blocklength 31, are contained in the cyclic Preparata code over Z₄of blocklength 31, which is defined in Section2.16and has the check matrix H =

1 ξ¹ ξ² · · · ξ³⁰  .

Likewise, the extended Calderbank–McGuire codes over Z₄, of blocklength 32, are contained in the extended Preparata code over Z₄of blocklength 32.

We do not provide detailed proofs of the minimum distances of the Calderbank–

McGuire codes here. Instead, we leave this as an exercise. Some methods of finding the minimum Lee distance of a code over Z₄are given in Section2.16. There we state that every codeword can be written as c(x) = c1(x)+2c2(x), where c1(x) and c2(x) have all coefficients equal to zero or one. Thus by reduction modulo 2, the Z₄polynomial c(x) can be dropped to the binary codeword c1(x). As a binary codeword, c1(x) has zeros atα¹,α³, andα⁵, and so has minimum Hamming weight at least equal to 7. If c₁(x) is zero, then c2(x) drops to a binary codeword with spectral zeros at α¹andα³. This means that c₂(x) has Hamming weight at least 5, so the Z4codeword 2c₂(x) has Lee weight at least 10. This codeword extends to a codeword with Lee weight at least equal to 12. For the second Calderbank–McGuire code, the codeword c(x) = 2c2(x) has Lee distance at least 14 and this codeword extends to a codeword with Lee weight at least 14. Other codewords of the Calderbank–McGuire code – those for which both c₁(x) and c2(x) are nonzero – are much harder to analyze.

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