2. STOPPING REDUNDANCY HIERARCHY BEYOND THE MINIMUM DISTANCE
2.1. Upper bounds on stopping redundancy
2.1.1. Upper bounds for general codes
In [46], Schwartz and Vardy presented an upper bound on the stopping redundancy of a general binary linear[n, k, d]codeC:
This bound is constructive. More precisely, the authors adjoin all linear combina-tions of up tod−2rows from the original parity-check matrix and prove that the resulting matrix has the stopping distanced.
The other related works are [50, 23, 22, 24, 25, 13, 19, 21], which present other constructive upper bounds—for general linear codes, for some specific families or for particular codes.
On the other hand, probabilisticarguments gave a rise to better bounds [22, 25, 19, 20, 54], yet these bounds are non-constructive. The main probabilistic technique in this thesis dates back to the work of Han and Siegel [19]. They established the following bound:
Briefly,En,d(t)is the average number of stopping sets of size at mostd−1in a parity-check matrix formed fromtdual codewords chosen randomly with repe-tition fromC⊥. Therefore, if for sometwe haveEn,d(t)<1, there is a realisation (i.e. choice oftdual codewords) when the obtained parity-check matrix has no stopping sets of size less thand. The term(r−d+ 1)is added to guarantee the correct rank of the obtained parity-check matrix.
The bound (2.2) has been improved by Han, Siegel, and Vardy in [20] by calculating probabilities in a more precise fashion and introducing one more stage of the probabilistic construction algorithm. At that stage, new rows are chosen one by one. We further refined this bound in [54] by carefully selecting the first non-random rows. This gives the smallest known values for most codes (to the
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best of our knowledge). A slightly modified version of the bound is presented in Theorem 19, which is the main result of this chapter.
Before proceeding, we prove the following technical result, which will be used further.
Lemma 18. For any integersi, j, r≥1, andj <2r, define π(r, i, j),1−i·2r−i
2r−j .
Then, for any integer r ≤ r0, and i ≤ i0, we have π(r, i, j) ≤ π(r0, i, j) and π(r, i, j)≤π(r, i0, j). In other words,π(r, i, j)is monotonically non-decreasing in integer variablesrandi.
Proof. The statement of the lemma follows easily if we rewrite:
π(r, i, j) = 1− i
2i · 1 1−j·2−r.
Below we present a bound modified from [54, Thm. 1]. More precisely, we drop the burdensome requirement
(r−1)(d−1)≤2d−1 (2.3)
thus making the bound applicable to all the binary linear codes. On the other hand, we need to add the rank deficiency term∆to ensure that the constructed parity-check matrix has the required rank. However, for medium and long codes, this term is negligible in comparison with the stopping redundancy.
Theorem 19. For an[n, k, d]linear binary codeCletH(τ)be anyτ ×nmatrix consisting ofτdifferent codewords of the dual codeC⊥and letuidenote the num-ber of stopping sets of sizei,i= 1,2, . . . , d−1, inH(τ). Fort= 0,1,· · · ,2r−τ, we introduce the following notations:
Dt=
d−1
X
i=1
ui
τ+t
Y
j=τ+1
π(r, i, j), Pt,0 =bDtc,
Pt,j = j
π(r, d−1, τ +t+j)Pt,j−1
k
, j = 1,2, . . .
∆ =r−max{rankH(τ), d−1}, and letκtbe the smallest integer such thatPt,κt = 0. Then
ρ≤τ+ min
0≤t<2r−τ{t+κt}+ ∆. (2.4)
Proof. We prove the theorem in two steps. First, we show the existence of a (τ +t)×nmatrix with a number of stopping sets less or equal toPt,0. Second, we show that this number further decreases when we add carefully selected rows one by one. Finally, after adding a sufficient number of rows, we obtain a matrix with no stopping sets of size less thand.
Step 1. By orthogonal array property, for any subset of columnsS ⊆ [n]of sizei,i= 1,2, . . . , d−1, there are exactlyi·2r−i codewords inC0⊥, that cover S. IfSis not covered byH(τ), none of thesei·2r−icodewords is present among the rows ofH(τ).
Fix a stopping set S in H(τ). Next, draw t codewords from the set C0⊥ \ {rows ofH(τ)}at random without repetition. There are
2r−τ −1 t
ways to do this, provided the order of selection does not matter. On the other hand, in the same setC0⊥\ {rows ofH(τ)}, there are(2r−τ−1)−i·2r−icodewords that donotcoverSand there are
(2r−τ −1)−i·2r−i t
ways to drawtcodewords out of them. Therefore, if we drawtcodewords from the setC0⊥\ {rows ofH(τ)}at random without repetition, the probability not to coverSby any one of them is
(2r−τ−1)−i·2r−i t
2r−τ −1 t
=
τ+t
Y
j=τ+1
π(r, i, j).
This holds for eachSthat was not originally covered byH(τ). Since the num-bers of the stopping sets of sizes1,2, . . . , d−1areu1, u2, . . . , ud−1, respectively, the average1 number of the stopping sets of size less than d that are left after adjoiningtrandom rows toH(τ)is
d−1
X
i=1
ui
τ+t
Y
j=τ+1
π(r, i, j),Dt.
Furthermore, since the above expression is an expected value of an integer random variable, there exists its realisation (i.e. choice oftrows), such that the number of stopping sets left is not more thanbDtc ,Pt,0. Fix thesetrows and further assume that we have a(τ +t)×nmatrixH(τ+t)with not more thanPt,0
stopping sets of size less thand.
1Averaging is by the choice oftrows.
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Step 2. Adjoin toH(τ+t)a random codeword fromC0⊥\ {rows ofH(τ+t)}. If some stopping setSof sizei,1≤i≤d−1, has not been covered byH(τ+t)yet, there are exactlyi·2r−i codewords inC0⊥\ {rows ofH(τ+t)}that cover Sand, thus, the probability thatSstays non-covered after adjoining this new row is
1− i·2r−i
2r−(τ +t+j) =π(r, i, τ +t+ 1)Lemma 18≤ π(r, d−1, τ +t+ 1).
This holds for any stopping setS of sizei. Then, there exists a codeword in C0⊥\ {rows ofH(τ+t)}such that after adjoining it as a row toH(τ+t), the number of non-covered stopping sets becomes less or equal to
j
π(r, d−1, τ +t+ 1)Pt,0k
,Pt,1.
To this end, we fix this new row and further assume that we have a(τ+t+1)×n matrixH(τ+t+1) with the number of the stopping sets of size smaller thandless or equal toPt,1. After that, we iteratively repeat Step 2. We stop when the number of non-covered stopping sets is equal to zero.
Finally, we need to ensure that the rank of the resulting matrix is indeed r.
We already know that it is not less thanrankH(τ). On the other hand, since we covered all the stopping sets of size less thand, the rank is at leastd−1. Hence it is enough to add∆additional rows to ensure the correct rank of the parity-check matrix.
Note. The expression in (2.4) is monotonically non-decreasing inui. Often, the exact values of ui are difficult to find and in that case upper bounds are used instead.
Note. By applying Lemma 18 to the expressions for DtandPt,j, we obtain that (2.4) is also monotonically non-decreasing inr. Sometimes, a parity-check mat-rix is redundant2 and the number of rows m is larger thanr. It might be more convenient to useminstead ofrand the bound (2.4) still holds.
To give a flavour of differences between the existing bounds on stopping re-dundancy, we calculate the bounds (2.1) in [46], (2.2) in [19], the bound in [20, Thm. 7], the bound in [54, Thm. 1], and the bound in Theorem 19. The two last bounds are calculated in two modes. First, we useτ = 1andH(τ)consists of the first row of the parity-check matrix of the corresponding code. Next, we use whole parity-check matrices of the codes asH(τ)(in Table 1,mdenotes the number of rows in a parity-check matrix used).
We calculate the aforementioned bounds for the following codes:
• the[24,12,8]extended Golay self-dual code (cf. Section 2.3.1);
• the [48,24,12] extended quadratic residue (QR) self-dual code (cf. [35, Sec. 16]);
2For instance, recall Gallager(J, K)-regular codes (cf. Section 1.2.3).
Table 1.Comparison of upper bounds on the stopping redundancy of different codes.
[24,12,8]
Golay
[48,24,12]
QR
[155,64,20]
Tanner
(2.1) 2509 4 540 385 6.2·1018
(2.2) 232 4440 1 526 972
[20, Thm. 7] 182 3564 1 260 673
[54, Thm. 1],τ = 1 180 3538 1 247 888
Theorem 19,τ = 1 185 3562 1 247 960
[54, Thm. 1],τ =m 168 2543 2573
Theorem 19,τ =m 168 2543 2573
• the(3,5)-regular[155,64,20]Tanner code in [49].
Table 1 presents numerical results. The original bound by Schwartz and Vardy (2.1) is the only constructive bound here, but it is by far the worst. Note that the bound in Theorem 19 is only slightly worse than [54, Thm. 1] but it is applicable to any code. Often, a code that do not satisfy (2.3) has its stopping distance equal to the minimum distance. Yet the new bound is useful for calculation of the stopping redundancy hierarchy (see Section 2.1.2).
The bounds in [54, Thm. 1] and Theorem 19 with τ = m give the tight-est results. However, they require knowledge of the stopping set spectrum of a parity-check matrix. For the Golay and the QR codes, we calculate their spectra by exhaustive brute-force checking. For the Tanner code, we use the spectrum obtained in [43, Tab. 1]. For longer codes, calculating a stopping sets spectrum can be infeasible even for the method in [43] and similar works. We suggest a way to overcome this obstacle in Section 2.2.3.