Abstract
Analysis of error correction performance for error correcting codes is very important when using such codes in digital communication systems. At medium-to-high signal-to-noise ratios, the distance spectrum of the error correcting code represents a good indicator for the error correction performance of the code. It is desired that the minimum distance of the code is as large as possible and that the corresponding multiplicity (i.e. the number of codewords having the weight equal to the minimum distance) is as small as possible. If we know an upper bound of the minimum distance of the code, then we have a good indication about the capabilities and the limitations of the code. One of the classes of the error correcting codes with the best performance is that of turbo codes. For such codes, establishing upper bounds on the minimum distance is challenging because it depends on the interleaver component of the turbo code. In this paper we consider turbo codes with component convolutional codes as in the Long Term Evolution standard. The interleaver lengths are of the form \(16 \varPsi \) or \(48 \varPsi \), with \( \varPsi \) a product of different prime numbers greater than three. The first achievement in the paper is that for these interleaver lengths, we show that cubic permutation polynomials (CPP), with some constraints on the coefficients, when \(3 \not \mid (p_i-1)\) for a prime \(p_i > 3\), always have a true inverse CPP. The most accurate upper bounds on the minimum distance for turbo codes are achieved by identifying bit information sequences leading to a certain weight of the corresponding turbo-codeword. In this paper we have indentified such bit information sequences by means of the full range dual impulse method to estimate the weight of the turbo-codewords. For the previously mentioned turbo codes and CPP interleavers, we show that the minimum distance is upper bounded by the values of 38, 36, and 28, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for quadratic PP (QPP) interleavers, the upper bound of the minimum distance is equal to 38. Several examples show that \(d_{min}\)-optimal CPP interleavers are better than \(d_{min}\)-optimal QPP interleavers because the multiplicities corresponding to the minimum distances for CPPs are about a half of those for QPPs. A theoretical explanation in terms of nonlinearity degrees for this result is given for all considered interleaver lengths and for the class of CPPs for which the upper bound is equal to 38.
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Acknowledgements
This work was supported by a National Research Grants - ARUT of the TUIASI, Project Number GnaC2018_39. The authors thank to the reviewers for for their helpful comments and suggestions which greatly improved this paper.
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Appendices
Appendix 1
Proof of Lemma 2:
\(\rho (x)\) is an inverse CPP of \(\pi (x)\) if
Taking into account Lemma 1, after some algebraic manipulations, Eq. (76) is equivalent to
Because \(\pi (x)\) and \(\rho (x)\) are true CPPs, from Lemma 1 it results that \(\rho _3 = f_3 = k_3 \cdot 2 \varPsi \), with \(k_3 \in \{ 1, 2, 3 \}\). Because \(p_i\) is odd \(\forall i \in \{ 1, 2, \dots , N_p \}\), \(\varPsi \) from (6) is also odd. Then, we can have \(\varPsi = 1 \ (\mathrm {mod}\ 8)\), \(\varPsi = 3 \ (\mathrm {mod}\ 8)\), \(\varPsi = 5 \ (\mathrm {mod}\ 8)\) or \(\varPsi = 7 \ (\mathrm {mod}\ 8)\). Then, \(2 \varPsi =2 \ (\mathrm {mod}\ 8)\) or \(2 \varPsi =6 \ (\mathrm {mod}\ 8)\). Because every \(p_i\) is odd and \(3 \not \mid p_i\), we can have \(\varPsi = 1 \ (\mathrm {mod}\ 24)\), \(\varPsi = 5 \ (\mathrm {mod}\ 24)\), \(\varPsi = 7 \ (\mathrm {mod}\ 24)\), \(\varPsi = 11 \ (\mathrm {mod}\ 24)\), \(\varPsi = 13 \ (\mathrm {mod}\ 24)\), \(\varPsi = 17 \ (\mathrm {mod}\ 24)\), \(\varPsi = 19 \ (\mathrm {mod}\ 24)\), or \(\varPsi = 23 \ (\mathrm {mod}\ 24)\). Then \(2\varPsi =2 \ (\mathrm {mod}\ 24)\), \(2\varPsi = 10 \ (\mathrm {mod}\ 24)\), \(2\varPsi =14 \ (\mathrm {mod}\ 24)\) or \(2\varPsi =22 \ (\mathrm {mod}\ 24)\).
Thus, for \(L = 16 \varPsi \) and \(\rho _2 = f_2 = k_2 \cdot 2 \varPsi \), with \(k_2 \in \{ 0, 1, 2, 3 \}\), (77) is equivalent to
For \(L = 16 \varPsi \), \(f_2 = k_2 \cdot 2 \varPsi \) and \(\rho _2 = ((k_2 +2) \ (\mathrm {mod}\ 4)) \cdot 2 \varPsi \), with \(k_2 \in \{ 0, 1, 2, 3 \}\), (77) is equivalent to
For \(L = 48 \varPsi \), \(\rho _2 = f_2 = k_2 \cdot 6 \varPsi \), with \(k_2 \in \{ 0, 1, 2, 3 \}\), (77) is equivalent to
For \(L = 48 \varPsi \), \(f_2 = k_2 \cdot 6 \varPsi \) and \(\rho _2 = ((k_2 + 2) \ (\mathrm {mod}\ 4)) \cdot 6 \varPsi \), with \(k_2 \in \{ 0, 1, 2, 3 \}\), (77) is equivalent to
Because \((2 \varPsi ) \mid L\), from (78), (79), (80), or (81), we have
Equation (82) is equivalent to
According to Theorem 57 from [34], we note that congruence \(f_1 \rho _1 = 2 \varPsi \cdot k + 1 \ (\mathrm {mod}\ L)\) has only one solution \(\rho _1\) modulo L when \(L=16 \varPsi \) or when \(L = 48 \varPsi \) and \(f_1 = 1 \ (\mathrm {mod}\ 3)\) or \(f_1 = 2 \ (\mathrm {mod}\ 3)\), because \(\gcd (f_1, L) = 1\). When \(L = 48 \varPsi \), with \( \varPsi = 1 \ (\mathrm {mod}\ 3)\), \(f_1 = 0 \ (\mathrm {mod}\ 3)\), and \(k \in \{1, 4, 7, \dots , 22 \}\), or when \(L = 48 \varPsi \), with \( \varPsi = 2 \ (\mathrm {mod}\ 3)\), \(f_1 = 0 \ (\mathrm {mod}\ 3)\), and \(k \in \{2, 5, 8, \dots , 23 \}\), congruence \(f_1 \rho _1 = 2 \varPsi \cdot k + 1 \ (\mathrm {mod}\ L)\) has three solutions modulo L because \(\gcd (f_1, L) = 3\) and \(3 \mid (2 \varPsi \cdot k + 1)\), but we will show that only the solution that fulfills condition \(\rho _1 = 0 \ (\mathrm {mod}\ 3)\) is valid and it is unique.
With (83), (78) is fulfilled if and only if
When \(2 \varPsi =2 \ (\mathrm {mod}\ 8)\), (84) is equivalent to
When \(2 \varPsi =6 \ (\mathrm {mod}\ 8)\), (84) is equivalent to
With (83), (79) is fulfilled if and only if
When \(2 \varPsi =2 \ (\mathrm {mod}\ 8)\), (87) is equivalent to
and when \(2 \varPsi =6 \ (\mathrm {mod}\ 8)\), (87) is equivalent to
Solutions \((k, \rho _1)\) of Eqs. (85), (86), (88), and (89) for each value of \(f_3 \in \{ 2\varPsi , 4\varPsi , 6\varPsi \}\), \(f_2 \in \{ 0, 2\varPsi , 4\varPsi , 6\varPsi \}\), and \(f_1 \ (\mathrm {mod}\ 8) \in \{ 1, 3, 5, 7 \}\), can be found using specific software programs. We have used symbolic calculus in Matlab for this goal. These solutions are unique for each value of \(f_1 \ (\mathrm {mod}\ 8)\) and they are given in Table 13, unified for the cases when \(2 \varPsi = 2 \ (\mathrm {mod}\ 8)\) and \(2 \varPsi = 6 \ (\mathrm {mod}\ 8)\).
With (83), (80) is fulfilled if and only if
When \(2 \varPsi =2 \ (\mathrm {mod}\ 24)\), (90) is equivalent to
When \(2 \varPsi =10 \ (\mathrm {mod}\ 24)\), (90) is equivalent to
When \(2 \varPsi =14 \ (\mathrm {mod}\ 24)\), (90) is equivalent to
When \(2 \varPsi =22 \ (\mathrm {mod}\ 24)\), (90) is equivalent to
With (83), (81) is fulfilled if and only if
When \(2 \varPsi =2 \ (\mathrm {mod}\ 24)\), (95) is equivalent to
When \(2 \varPsi =10 \ (\mathrm {mod}\ 24)\), (95) is equivalent to
When \(2 \varPsi =14 \ (\mathrm {mod}\ 24)\), (95) is equivalent to
When \(2 \varPsi =22 \ (\mathrm {mod}\ 24)\), (95) is equivalent to
Solutions \((k, \rho _1)\) of equations (91)–(94) and (96)–(99) for each value of \(f_3 \in \{ 2\varPsi , 4\varPsi , 6\varPsi \}\), \(f_2 \in \{ 0, 6\varPsi , 12\varPsi ,\) \( 18\varPsi \}\), and \(f_1 \ (\mathrm {mod}\ 24) \in \{ 1, 3, \dots , 23 \}\), found by software means, are given in Tables 14, 15, 16, 17 and 18. When the congruence equation \(f_1 \rho _1 = 2 \varPsi \cdot k +1 \ (\mathrm {mod}\ 48 \varPsi )\) has three solutions in variable \(\rho _1 \ (\mathrm {mod}\ 48 \varPsi )\), the valid solution (i.e. that fulfills one of the equations (91)–(94)) is only \(\rho _1 = f_1 \ (\mathrm {mod}\ 24)\). We note that, because of condition \((f_1 + f_3)\ne 0 \ (\mathrm {mod}\ 3)\) for \(L = 48 \varPsi \), for a certain value of \(f_3\), \(f_1 \ (\mathrm {mod}\ 24)\) can take only 8 different values from the 12 ones from the set \(\{ 1, 3, \dots , 23 \}\). For example, if \(f_3 = 2 \varPsi \) and \(k_{ \varPsi } = (2 \varPsi ) \ (\mathrm {mod}\ 24) = 2\), then \(f_3 = 2 \ (\mathrm {mod}\ 3)\) and \(f_1 \ (\mathrm {mod}\ 24) \in \{ 3, 5, 9, 11, 15, 17, 21, 23 \}\). \(\square \)
Appendix 2
According to the results from [33], the nonlinearity degree of CPP interleavers of even lengths is equal to
where \(N_{k_0,QNP_1}\) is the number of the common solutions \(k_0\) of congruence equations
For coefficients given in Table 7, we have
Because \(\gcd ( 3 f_3 , L) \mid (L/2)\), the solutions of the first equation from (101) are of the form
where \(k_{0,f3}\) is equal to
Thus, the solutions of the first equation from (101) are
Similarly, we have
Because \(\gcd ( 2 f_2 + L/2 , L) \mid (L/2)\) only for \(k_2 \in \{ 0, 1, 3 \}\), the second equation from (101) has solutions only for these three values of \(k_2\). These solutions are of the form
where \(k_{0,f_2}\) is equal to
Thus, the solutions of the second equation from (101) are
Because
and
it results that the two equations from (101) have no common solutions, i.e. \(N_{k_0,QNP_1} = 0\). Because for \(k_3 \in \{ 1, 3 \}\)
from (100) we have
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Trifina, L., Tarniceriu, D., Ryu, J. et al. Upper bounds on the minimum distance for turbo codes using CPP interleavers. Telecommun Syst 76, 423–447 (2021). https://doi.org/10.1007/s11235-020-00723-4
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DOI: https://doi.org/10.1007/s11235-020-00723-4