Distance Bounds for Periodically Time-Varying and Tail-Biting LDPC Convolutional Codes

Existence type lower bounds on the free distance of periodically time-varying LDPC convolutional codes and on the minimum distance of tail-biting LDPC convolutional codes are derived. It is demonstrated that the bound on free distance of periodically time-varying LDPC convolutional codes approaches the bound on free distance of general (nonperiodic) time-varying LDPC convolutional codes as the period increases. The proof of the bound is based on lower bounding the minimum distance of corresponding tail-biting LDPC convolutional codes, which is of interest in its own right.

presented 1 in [3].While minimum distance bounds for block LDPCs were derived in Gallager's original work [2], the first analytical lower bound on the free distance of LDPCCCs was only derived recently [1].The proof presented in [1] holds for an ensemble of general (non-periodic) time-varying LDPCCCs and must employ a special expurgation technique to compensate for the non-periodic structure of the ensemble.
In this paper, we derive an existence type lower bound on the free distance of periodically timevarying LDPCCCs.We show that, as the period increases, the new bound approaches the bound on free distance of non-periodic LDPCCCs derived in [1].The proof presented for the new bound is based on considering the minimum distance of tail-biting LDPCCCs (TB-LDPCCCs) [8].
In particular, we lower bound the minimum distance of TB-LDPCCCs constructed from an ensemble of periodically time-varying LDPCCCs and use this to lower bound the free distance of the original ensemble.
Tail-biting was introduced by Solomon and van Tilborg [9] and independently by Ma and Wolf [10] as a method of terminating a convolutional code without the rate loss caused by standard termination.The resulting tail-biting codes have a dual nature, i.e., they simultaneously have the properties of both block and convolutional codes.As a consequence, their minimum distance depends both on the block length of the tail-biting code and the constraint length of the convolutional code.
The minimum distance of conventional (non-LDPC) tail-biting codes equals the minimum of two related distance measures, d intra and d inter [11].The intra minimum distance d intra reflects the convolutional code properties of the tail-biting code and is lower bounded by the Costello bound [6] on the free distance of convolutional codes.The inter minimum distance d inter reflects the block code properties of the tail-biting code and is lower bounded by the Varshamov-Gilbert bound [4], [5] on the minimum distance of block codes.Analogous to conventional tail-biting convolutional codes, the minimum distance of TB-LDPCCCs is lower bounded by the minimum of d intra and d inter , where d intra is lower bounded by the bound on free distance of LDPCCCs derived in [1] and d inter is lower bounded by Gallager's bound on minimum distance of LDPC block codes [2].
The paper is organized as follows.Section II presents the definition of the LDPCCC code ensemble considered.Section III is devoted to lower bounding the minimum distance of TB-LDPCCCs, and a lower bound on the free distance of periodically time-varying LDPCCCs is proved in Section IV.Numerical results are given in Section V, and Section VI concludes the paper.

II. AN LDPC CONVOLUTIONAL CODE ENSEMBLE
In [3], a rate R = b/c binary convolutional code was defined as the set of sequences where the semi-infinite syndrome former (transposed parity-check) matrix H T [0,∞] is given by and each entry To satisfy an easy encoding property (see [3], [13]), the matrices H T 0 (t) must have full rank for all time instants t, and hence we assume that the last (c − b) rows of H T 0 (t) are linearly independent for all t.Then the first b symbols of v t at each time instant t are information symbols and the last (c−b) symbols are parity symbols.The largest i such that H T i (t+i) is a non-zero matrix for some t is called the syndrome former memory m s .A (J , K ) regular LDPCCC is defined by a syndrome former that contains exactly J ones in each row and K ones in each column (starting from the ((c − b)m s + 1)th column).Now we define a special sub-class of (J, K) regular LDPCCCs, where the component submatrices H T i (t) are composed of M × M binary permutation matrices 2 .Let a = gcd(J, K) denote the greatest common divisor of J and K. Then there exist positive integers J ′ and K ′ such that J = aJ ′ and K = aK ′ and gcd(J ′ , K ′ ) = 1.For i = 0, 1, . . ., a − 1, the K ′ M × J ′ M sub-matrices H T i (t + i) of the syndrome former are where each P (k,j) i (t + i), k = 0, 1, . . ., K ′ − 1, j = 0, 1, . . ., J ′ − 1, is an M × M permutation matrix.All other entries of the syndrome former are K ′ M × J ′ M zero matrices.We assume that the matrix H T [0,∞] is periodically time-varying with period T , i.e., In this case, a code is characterized by a section H T [0,T −1] of the semi-infinite syndrome former for a periodically time-varying (3,6) regular LDPCCC is shown in Fig. 1.In this case, the code construction parameters are K = 6, J = 3, a = 3, c = 2M, and b = M.Each matrix H T i (t + i) consists of two M × M permutation matrices, i.e., where (4) has full rank equal to M. Therefore the code rate is M/2M.Note that by permuting rows of the syndrome former, an equivalent rate 1/2 (3, 6) regular LDPCCC with syndrome former memory at most 3M − 1 can be obtained (see [1]).Now suppose that the M × M permutation matrices comprising the sub-matrices (2) of the syndrome former H T [0,T −1] are chosen independently and such that each of the M! possible permutation matrices is equally likely.Then we obtain a random ensemble of (J , K ) regular T -periodic LDPCCCs, which we designate C(J, K, M, T ).
The syndrome formers in the ensemble C(J, K, M, T ) have memory m s = a − 1, independent of M, while b and c depend on M.This ensemble of codes is different from the LDPCCCs considered in [3], [12], and [13], where the codes have varying syndrome former memories m s , while the rate parameters b and c are fixed.For the ensemble C(J, K, M, T ), as M increases, i.e., as b and c increase, the syndrome formers become increasingly sparse.
During the encoding process, the information sequences are divided into blocks of b = (K ′ − J ′ )M symbols, which are input to an LDPC convolutional encoder at each time instant t, and a block of c = K ′ M encoded symbols is generated at the output.For any code in C(J, K, M, T ), an equivalent systematic LDPC convolutional encoder can be constructed such that the computational complexity per encoded parity-check symbol depends only on K and is independent of the permutation matrix size M (see [3]).
Since there are at least J ′ linearly dependent columns in H T 0 (t) for any code in C(J, K, M, T ), code.The constraint length of codes from C(J, K, M, T ) For example, the codes in the ensemble For T ≥ J, a syndrome former HT [0,T −1] for a (J , K ) regular TB-LDPCCC can be constructed from one period of a syndrome former H T [0,T −1] for a (J, K) regular T -periodic LDPCCC.This can be done by wrapping back the last J − 1 blocks of columns of (see Fig. 1) for any T ≥ 3, as illustrated in Fig. 2. TB-LDPCCC's created in this way form an ensemble which we denote by C(J, K, M, T ).The block length of these codes is 2T M. In the following section we will use this ensemble to derive a lower bound on the minimum distance of TB-LDPCCCs.
In the following section, we show how this existence bound on the minimum distance of TB-LDPCCCs leads to an existence bound on the free distance of the T -periodic LDPCCCs.

IV. A LOWER BOUND ON THE FREE DISTANCE OF T -PERIODIC LDPCCCS
We begin by considering TB-LDPCCCs of length 2κT M, where κ is an integer satisfying where the transposed parity-check matrix HT [0,κT −1] of the TB-LDPCCC is constructed from a syndrome former H T [0,κT −1] of a T -periodic LDPCCC by wrapping back the last two blocks of columns (see Section II).The product of a codeword ṽ[0,κT−1] in the length 2κT M TB-LDPCCC and the syndrome former H T [0,κT −1] of the T -periodic LDPCCC defines an M(κT +2)dimensional syndrome vector where the syndrome vector is a concatenation of M-dimensional subvectors s t = (s t1 , s t2 , . . ., s tM ), t = 0, 1, . . ., κT + 1.
Since ṽ[0,κT−1] satisfies (20) and HT [0,κT −1] is constructed from H T [0,κT −1] using the wrapping back procedure of Fig. 2, the subvectors s t satisfy the conditions and Proof: The proof follows from the definitions of free distance and row distance.The Lth order row distance d r L of a periodically time-varying convolutional code is defined [11] as the minimum weight of all code sequences having a nonzero segment of length at most L + m + 1 (in this case, the code sequences are composed of blocks of length 2M), where m is the encoder memory.In turn, the free distance d free is defined as where d r L is monotonically decreasing with L and there exists an integer L 0 such that for any L > L 0 .
Thus, we can find a sufficiently large κ 0 such that, for any κ ≥ κ 0 , the code sequences of the length 2κT M tail-biting code include all possible nonzero segments of length L 0 + m + 1 blocks of the T -periodic convolutional code.This implies that which, along with (28), leads to (26).
Now let ṽ[0,κT−1] = (ṽ 1) , . . ., ṽ(0) κT −1 , ṽ κT −1 ) be a codeword in the length 2κT M TB-LDPCCC, i.e., it satisfies (20).Note that this codeword can be represented as ṽ where ṽ[(i−1)T,iT−1] = (ṽ Then consider the sequence i.e., the modulo-2 sum of the components of the codeword ṽ[0,κT−1] given in (30).The following lemma proves that v[0,T−1] is a codeword in the TB-LDPCCC consisting of only one period of the T -periodic LDPCCC.The distance ratios for the various LDPCCC ensembles are presented in Fig 4. We see that the minimum distance to block length ratio for TB-LDPCCCs is equal to Gallager's ratio for LDPC block codes when the period T is small.For larger periods, however, the ratio drops and tends to zero, due to the effect of d intra , as noted above.On the other hand, the free distance to constraint length ratio for T -periodic LDPCCCs grows with increasing T and approaches the ratio derived in [1] for general (non-periodic) time-varying LDPCCCs as T increases beyond 11.

VI. CONCLUSIONS
In this paper, we derived a lower bound on the free distance of periodically time-varying (J, K) regular LDPCCCs and a lower bound on the minimum distance of the associated TB-LDPCCCs.Theorems 1 and 2 give analytical expressions for these bounds in the general case.
Using these expressions, we calculated numerically the bounds on free distance and minimum distance for the practically interesting (3, 6) regular LDPCCC case.In the limiting cases, for T > 11 the free distance bound corresponds to the bound for general (non-periodic) time-varying LDPCCCs derived in [1], and for T = 3 the minimum distance bound corresponds to Gallager's bound for LDPC block codes.

Fig. 1 .
Fig. 1.One period of a syndrome former for a code in the ensemble C(3, 6, M, T ).

Fig. 2 .
Fig. 2. Syndrome former of a TB-LDPCCC in the ensemble C(J, K, M, T )

t
vectors ṽ[0,T−1] with weight composition d[0,T−1] .Our goal is to calculate the average number of codewords ṽ[0,T−1] with weight composition d[0,T−1] for a code in the ensemble C(3, 6, M, T ).Finally, in the asymptotic January 5, 2008 DRAFT case, as M → ∞, it is more convenient to operate with the normalized weight composition ρ

Figs. 3
Figs.3 (b) and 3 (c), the components of the maximizing vector are no longer approximately equal in these cases), and further increases in the period do not lead to higher values of this sum.It follows that the normalized weight drops as T increases beyond 11.This effect is observed due to the intra minimum distance d intra , which is lower bounded by the bound on free distance of LDPCCCs derived in[1].This bound scales as α LDPCCC(3,6)ν = 6α LDPCCC (3, 6)M ≈ 0.5Mand represents an upper bound on the minimum distance of TB-LDPCCCs.In other words, the minimum distance to block length ratio of TB-LDPCCCs decreases as T increases beyond 11, since the block length continues to increase while the minimum distance cannot grow beyond the constant d intra .

Fig. 4 .
Fig. 4. Distance ratios for T -periodic LDPCCCs and TB-LDPCCCs as a function of the period T .