Optimal PGMs with the lowest McMillan degree

In previous section, we know that there exists an optimal polynomial generator matrices, but we further want to know how to find one with the lowest McMillan degree. Accord-ing to the ideas in [8], we can scratch the procedure roughly. By the method described in Section 2.3, we can obtain an optimal and basic generator matrix G(D) for any convolu-tional code C. By Theorem 1 and Theorem 7, we know that if T (D) is unimodular and effectively lower-triangular matrix with respect to G(D), then T (D)G(D) is optimal and intdeg(G(D))=intdeg(T (D)G(D)). Given a generator matrix G(D), its McMillan degree must larger or equal to its internal degree. Since G(D) is basic, it has the lowest internal degree. So we start from the internal degree of a basic PGM G(D), that is, the possible minimum McMillan degree for this code, we check whether there is a G⁰(D) whose McMillan degree is equal to the internal degree when we fix a internal degree value. Moreover, by Corollary 1, we know that Mcdeg(G(D))=Pk

i=1(m_i − m_i−1)⁺, where m_i is the maximum degree of all i × i minors. It imimplies that Mcdeg(G(D))≥ m₁, that is, Mcdeg(G(D)) is not less than the degree of all entries of G(D).

Let m_i(G(D)) be m_i of G(D), where m_i is the maximum degree of i × i minors. Suppose a k × n generator matrix G(D) with intdeg(G(D))=κ, and let G_T(D) = T (D)G(D) where T (D) is a unimodular and effectively lower-triangular matrix with respect to G(D), then intdeg(G_T(D))=κ. If m₁ of G_T(D) is greater than κ, it implies that

M cdeg(G_T(D)) ≥ m₁(G_T(D)) > κ = intdeg(G_T(D)),

that is, Mcdeg(G_T(D))¿intdeg(G_T(D)). Since we work with the finite field F , suppose F = GF (q), we know that the numbers of G_T(D) with m₁(G_T(D)) < κ at most q^κnkbecause every element of G_T(D) is of the form q₀+ q₁D + ... + q_κD^κ, where q_i ∈ F , ∀1 ≤ i ≤ k. If two matrices T₁(D) and T₂(D) such that T₁(D)G(D) = T₂(D)G(D) = G_T(D) has m₁ < κ,

it implies T₁(D) = T₂(D). Hence there are finite number matrices T (D) such that m₁ of G_T(D) is less or equal to κ. Among all these matrices T (D), if there is no any T (D) such that G_T(D) = T (D)G(D) with m_k ≥ m_k−1, where m_k and m_k−1 means the maximum degree of k × k minors and (k − 1) × (k − 1) minors of G_T(D) respectively, it means that there is no such a PGM with McMillan degree equal to internal degree, i.e. no minimal and optimal PGMs. Then we multiply G(D) by A(D) to increase internal degree by 1, where A(D) is a k × k diagonal matrix of the following form:

A(D) = and let G⁰_T0(D) = T⁰(D)G(D), where T⁰(D) is unimodular and effectively lower-triangular with respect to G⁰(D). Similarly, the number of G⁰_T0(D) with m₁(G⁰_T0(D)) < κ + 1 at on that add 1 to internal degree and do the same thing again. Since we can not find any optimal generator matrix with McMillan degree equal to κ, so we know that G⁰_T0(D) is the desired optimal PGM with the lowest McMillan degree.

There are a lot of unimodular and effectively lower-triangular matrices T (D) with respect to G(D), so we discuss more detail about these matrices to reduce the number of T (D) we need to check hence reduce the complexity of this procedure. If G(D) is a generator matrix for a given convolutional code C, T (D) is any k × k nonsingular matrix, then T (D)G(D) will span the same codeword space as the codeword space G(D) spans. Also, a nonsingular matrix T (D) is composed of three types of elementary matrices. First elementary operation

is interchanging two row, and this operation does not change any k × k and (k − 1) × (k − 1) minors so that does not change m_k(G(D)) and m_k−1(G(D)). If an optimal PGM G(D) with m_k(G(D)) < m_k−1(G(D)), it is not the desired PGM we want to find. Suppose we do the first operation on G(D), the operation has no influence on its m_k(G(D)) and m_k−1(G(D)), that is, for the new PGM T (D)G(D), where T (D) is a matrix make two rows of G(D) exchange, m_k(T (D)G(D)) is still less than m_k−1(G(D)), then T (D)G(D) is also not the desired matrix. Due to the former cause, we will deduct Type 1 operation when we multiply G(D) by a nonsingular matrix. The second elementary row operation is multiplying a row with a polynomial α(D), this operation will increase the internal degree of G(D) by degree of determinant of α(D), so we will not execute this operation when fixing a internal degree value. The last operation, adding a polynomial multiple of a row to another row, does also not change m_k, but it may change m_k−1, that is, m_k−1(T (D)G(D)) may be smaller and equal than m_k−1(G(D)), where T (D) is the matrix corresponding to Type 3 operation.

According the above discussion, we now give a way to find an optimal polynomial gener-ator matrix with the fewest McMillan degree. Given a convolutional code C ,by Procedure 1, we can obtain an optimal generator matrix, then we transform it to an optimal and basic PGM G_o(D) by using the method introduced in Section 2.3. Let intdeg(G_o(D))=κ.

As we mentioned before, we know that any PGM G(D) has Mcdeg(G(D))≥ m₁, where m₁ is the maximum degree of 1 × 1 minors of G(D). Hence if there is a uni matrix T (D) such that m₁(T (D)G_o(D)) is greater than κ, then Mcdeg(T (D)G_o(D))>κ. Let Π = {∀U (D) : m₁(U (D)G_o(D)) ≤ κ}, where U (D) is an unimodular and effectively lower triangular with respect to G_o(D). According to the above discussion, U (D) is only com-posed of Type 3 operation. Since we work with finite field F , as we explained before, there are finite number of matrices in Π. By the previous corollary, we know that for a PGM G(D), intdeg(G(D))≤Mcdeg(G(D)), the equality holds when m_k(G(D)) ≥ m_k−1(G(D)).

Hence if there is a matrix U_el(D) ∈ Π such that m_k(U_elG_o(D)) ≥ m_k−1(U_elG_o(D)), it implies that Mcdeg(U_elG_o(D))=intdeg(U_elG_o(D))=κ, and hence it is minimal and optimal.

If there is no such a PGM U_elG_o(D) that m_k(U_elG_o(D)) ≥ m_k−1(U_elG_o(D)), it means that

there are no PGMs which is optimal and minimal. Then let G_o(D) be multiplied by A(D), where A(D) is a k × k diagonal matrix as follows:

 unimod-ular and effectively lower-triangunimod-ular matrix which is only composed of Type 3 operation, that is, adding a polynomial multiple of a row to another row. Similarly, there are fi-nite number of matrices in Π_A, if there exists a generator matrix U_elAG_A(D) such that m_k(U_elAG_A(D)) ≥ m_k−1(U_elAG_A(D)), then it is an optimal PGM with the lowest McMillan degree κ + 1; else let G(D) be multiplied by A(D) again, where A(D) is of the above form with the sum of degree of diagonal term is equal to 2, and recursive searching again. Now, we conclude this as below:

Procedure 2.

Step 1 Given an (n, k) convolutional code C, by Procedure 1 and later transformation, we obtain an optimal and basic generator matrix G(D), intdeg(G(D))=κ,

s(G(D)) = (s₁, ..., s₁

where w₀+ w₁+ ... + w_k−1 = 1.

Step 3 Choose an element from Σ_i, named A_c(D). Let Σ_i = Σ_i\A_c(D). If m_k(A_c(D)G(D)) ≥ m_k−1(A_c(D)G(D)), go to Step 5; otherwise, let Ω be a set consisting of all (k − 1) × (k − 1) submatrices in A_c(D)G(D) that satisfy

deg(det(M_i,ji(D)(D)) = m_k−1(A_c(D)G(D)),

where M_i,ji(D), 1 ≤ i ≤ |Ω|, denotes i-th element in Ω and the entries in M_i,ji(D) are collected from all rows except for the j_i-th row in A_c(D)G(D). Let t = |Ω| and Γ = St

i=1{j_i}. Let Γ⁰ = 1, 2, ..., k \ Γ and Ψ = {U (D) : U (D) is U-ELT, i-th column of U (D) is a unit vector e_i, ∀i ∈ Γ⁰ and m₁(U (D)A_c(D)G(D))≤ κ + i}. Check all U⁰(D)A_c(D)G(D) with U⁰(D) ∈ Ψ. If there is a PGM and m_k(U⁰(D)A_c(D)G(D)) ≥ m_k−1(U⁰(D)A_c(D)G(D)), go to Step 5; else go to the next step.

Step 4 Check if Σ_i is empty. If yes, set i = i + 1 and go to Step 2; otherwise, go back to Step 3.

Step 5 Set G^∗(D) = U⁰(D)A_c(D)G(D) and it is an optimal PGM with the lowest McMillan degree κ + i.

Example 2. Suppose a convolutional code C can be generated by an basic and optimal generator matrix G(D) as

G(D) =









1 0 D⁵ 0

1 1 D³ 1 + D + D² 1 1 1 + D³ D²









, where s(G(D)) = (2, 3, 3) and m₃=intdeg(G(D))=6. By calculating its all 2 × 2 minors, we get m₂ = 7 > m₃, so intdeg(G(D))6= Mcdeg(G(D)). Let A(D) = I. Since the numbers of repetitions of separation values are 1 and 2 and the first 2 × 2 submatrix with degree equals to 7 is constructed by 1-st and 2-nd rows and second is constructed by 1-st and 3-rd rows, we fill out 1-st column by an unit vector and candidate for unimodular and effectively

lower-triangular U (D) are of the form: optimal PGM with lowest McMillan degree 6.

Chapter 4

Optimal generator matrices with the