# New bounds and constructions for constant weighted X-codes

As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic values (Xs). From a combinatorial view point, Fujiwara and Colbourn FC2010 introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we prove a general lower bound on the maximum number of codewords n for an (m,n,d,x) X-code of weight w, and we further improve this lower bound for the case with x=2 and w=3 through a probabilistic hypergraph independent set approach. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted X-codes with d=3,5 and x=2, which are nearly optimal for the case when d=3 and w=3. We also consider a special class of X-codes introduced in FC2010 and improve the best known lower bound on the maximum number of codewords for this kind of X-codes.

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## I Introduction

Typical digital circuit testing applies test patterns to the circuit and observes the circuit’s responses to the applied patterns. The observed responses are compared to a test pattern with expected responses, and a chip in the circuit is determined to be defective if the comparison mismatches. With the development of the large scale integrated circuits (IC), although the comparison for each testing output is simple, the ever increasing amount of testing data costs much more time and space for processing. This leads to the requirement of more advanced test compression techniques [27]. Since then, various related techniques have been studied such as automatic test pattern generation (ATPG) (see [21, 30, 15, 6] and the reference therein) and compression-based approaches (e.g., [28, 26]). The technique of -compact is one of the compression-based approaches that have high error detection ability in actual digital systems [26].

Usually, test engineers obtain the expected responses through fault-free simulations of the circuit for the applied test patterns. But fault-free simulations cannot always determine the expected responses. In some cases, due to uninitialized memory elements, bus contention, inaccurate simulation models, etc (see Table 2 in [26]), the responses may contain unknown logic values. These unknown bits are denoted by s, and the idea of -compact provides a technique for reliable test response compaction in the presence of s [25].

-compact uses -codes as linear maps to compress test responses. An -code is an

binary matrix with column vectors as its codewords. The parameters

correspond to the test quality of the code. The weight of a codeword is the number of s in . The value of is called the compaction ratio and -codes with large compaction ratios are desirable for actual IC testing.

For -codes of arbitrary weight, let be the maximum number of codewords for which there exists an -code. In [13], based on a combinatorial approach, Fujiwara and Colbourn obtained a general lower bound on using probabilistic method (see Theorem 4.6, [13]). And this lower bound was further improved to by Tsuboda et al. in [36].

For constant weighted -codes, let be the maximum number of codewords for which there exists an -code of constant weight . Since factors like power requirements, compactor delay and wireability require the weight of each codeword to be small to meet the practical limitations (see [26, 37]) and codewords with weight at most are not essential when considering the compaction ratio (see [23, 13]). Therefore, aiming to achieve a large compaction ratio while minimizing the weight of each codeword, many works have been done about -codes of constant weight .

In [37], by viewing the matrix of an -code as an incidence matrix of a graph, Wohl and Huisman build a connection between -codes of constant weight and graphs with girth at least . For the cases with multiple ’s, using results from combinatorial design theory and superimposed codes, Fujiwara and Colbourn [13] proved that and . And they studied a special class of -codes of constant weight with a property that boosts test quality when there are fewer unknowable bits than anticipated. Recently, Tsunoda and Fujiwara [35] proved that for and they also improved the lower bound on the maximum number of codewords for the above special class of -codes of constant weight introduced in [13].

In this paper, we focus on the constant weighted -codes. Based on the results from additive combinatorics and extremal graph (hypergraph) theory, we obtain the following results:

• A general lower bound for constant weighted -codes:

 Mw(m,d,x)≥(1−o(1))(m⌈w/(x+d−1)⌉)(w⌈w/(x+d−1)⌉).
• An improved lower bound for -codes of constant weight with for any :

 M3(m,d,2)≥c⋅m76(logm)16,

for some absolute constant .

• Explicit constructions for constant weighted -codes with and . These constructions further improve the general lower bound and obtain nearly optimal lower bounds for the cases and , when is large enough.

• An improvement of of the best known lower bound on the maximum number of codewords for the special class of -codes of constant weight introduced in [13]. This improvement is also extended to the general case where higher error tolerances are required.

This paper is organised as follows: In Section II, we introduce the formal definitions for -codes and superimposed codes, we also include a well-known lower bound for hypergraph independent sets. In Section III, we investigate the bounds and constructions for constant weighted -codes. We prove a general lower bound on and a non-trivial lower bound on . We also present some explicit constructions for constant weighted -codes with and based on the results from additive combinatorics and finite fields. In Section IV, we improve the lower bound on the maximum number of codewords for a special class of -codes of constant weight and extend this result to a general case. In Section V, we conclude our work with some remarks.

## Ii Preliminaries

### Ii-a Notation

We use the following notations throughout this paper.

• Let be the power of a prime , be the finite field with elements, be the vector space of dimension over .

• For any vector , let and For a set define , where for and .

• For any integer , denote as the set of the first consecutive positive integers .

• For simplicity, denote AP as the shortened form of arithmetic progression and -AP as the shortened form of arithmetic progression of length .

### Ii-B X-Codes

Consider two -dimensional vectors and where . The addition of and is bit-by-bit addition, denoted by ; that is

 s1⊕s2=(s(1)1+s(2)1,s(1)2+s(2)2,…,s(1)m+s(2)m).

The superimposed sum of and , denoted by , is

 s1∨s2=(s(1)1∨s(2)1,s(1)2∨s(2)2,…,s(1)m∨s(2)m),

where if , otherwise . And we say an -dimensional vector covers an -dimensional vector if .

For a finite set of -dimensional vectors, define

 ⨁S=s1⊕⋯⊕ss,

and

 ⋁S=s1∨⋯∨ss.

When , , and when , define (i.e. the zero vector).

###### Definition II.1.

[13] Let be a positive integer and a nonnegative integer. An -code is a set of -dimensional vectors over such that and

 (⋁S1)∨(⨁S2)≠⋁S1 (1)

for any pair of mutually disjoint subsets and of with and . A vector is called a codeword. The weight of the code word is . The ratio is called the compaction ratio of .

Let be the maximum number of codewords for which there exists an -code. From the definition above, when , the codewords of an -code actually form an parity check matrix of a binary linear code of length with minimum distance . Therefore, -codes can be viewed as a special kind of traditional error-correcting codes.

For the case when and , as pointed out in [23], an -code is equivalent to a -superimposed code of size .

###### Definition II.2.

[19] A -superimposed code of size is an matrix with entries in such that no superimposed sum of any columns of covers any other column of .

Superimposed codes are also called cover-free families and disjunct matrices. These kinds of structures have been extensively studied in information theory, combinatorics and group testing. Thus, the bounds and constructions of -superimposed codes can also be regarded as those for -codes (see, for example, [5, 8, 9, 10, 14, 16, 33]).

When and , according to the definition, an -code is also an -code and an -code. From [23], an -code is an -code as well.

Given an -code, Fujiwara and Colbourn [13] showed that a codeword of weight less than or equal to does not essentially contribute to the compaction ratio (see also [23]). Therefore, when considering -codes of constant weight , we always assume that .

### Ii-C Independent sets in hypergraphs

A hypergraph is a pair , where is a finite set and is a family of subsets of . The elements of are called vertices and the subsets in are called hyperedges. We call a -uniform hypergraph, if all the hyperedges have the same size , i.e., . For any vertex , we define the degree of to be the number of hyperedges containing , denoted by . The maximum of the degrees of all the vertices is called the maximum degree of and denoted by .

An independent set of a hypergraph is a set of vertices containing no hyperedges and the independence number of a hypergraph is the size of its largest independent set. There are many results on the independence number of hypergraphs obtained through different methods (see [2], [3], [7], [18]). Recall that a hypergraph is linear if every pair of distinct hyperedges from intersects in at most one vertex. In this paper, we shall use the following version of the famous result of Ajtai et al. [2] due to Duke et al. [7] to derive some lower bounds on .

###### Lemma II.3.

[7] Let and let be a -uniform hypergraph with . If is linear, then

 α(H)≥c⋅v(H)⋅(logDD)1k−1, (2)

for some constant that depends only on .

## Iii Bounds and constructions of constant weighted X-codes

In this section, we consider -codes of constant weight. This section is divided into four subsections. Section III-A includes some known results and a general upper bound on the number of codewords of constant weighted -codes. In Section III-B, based on a result of packing, we obtain a general lower bound on the maximum number of codewords of constant weighted -codes. Then in Section III-C, we give some explicit constructions for constant weighted -codes with and . And in Section III.D, we improve the general lower bound for -codes of constant weight with .

### Iii-a A general upper bound and known results

Denote as the maximum number of codewords of an -code of constant weight . Since the restrictions for -codes get more rigid with the growing of , thus we have

 Mw(m,d,x)≤Mw(m,1,x). (3)

In 1985, Erdös et al. [10] proved the following bounds on the maximum number of codewords of a -superimposed code of constant weight .

###### Theorem III.1.

[10] Denote as the maximum number of columns of a -superimposed code of constant weight . Let . Then, we have

 (mt)(wt)2≤fx(m,w)≤(mt)(w−1t−1).

Moreover, if we take where , then for ,

 fx(m,w)≥(1−o(1))(m−δt)(w−δt),

and holds in the following cases:1) ; 2) ; 3) and . Moreover, equality of the latter upper bound holds if and only if there exists a Steiner -design .

By the equivalency between an -code and a -superimposed code, we have the following immediate consequence:

###### Theorem III.2.

Let where . Then, for all ,

 Mw(m,d,x)≤Mw(m,1,x)≤(mt)(w−1t−1). (4)

And for ,

 Mw(m,d,x)≤Mw(m,1,x)≤(m−dt)(w−dt) (5)

holds in the following cases:1) ; 2) ; 3) and .

In particular, for the case , Theorem III.2 actually gives the following upper bound

 Mw(m,d,2)≤⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(m−1w/2)(w−1w/2),when w is even;(m(w+1)/2)(w(w+1)/2),when w is odd% . (6)

According to the results from design theory, Fujiwara and Colbourn [13] proved the above upper bound is tight for the case and , when there exists a corresponding Steiner triple system. Using the well-known graph removal lemma, Tsunoda and Fujiwara [35] improved this upper bound on to for . So far as we know, for and , no upper or lower bounds better than these can be found in the literature.

### Iii-B A general lower bound from maximum (t,w,m)-packings

Let be an -element set. A is called a -packing if holds for every pair . In [29], V. Rödl proved the following lower bound on the size of the maximum -packing.

###### Theorem III.3.

[29] For all positive integers , there exists a constant , such that when , we have

 max{|P:P~{}is a~{}(t,w,m)-packing|}=(1−o(1))(mt)(wt). (7)

Based on the above result for maximum -packings, we can obtain the following asymptotic general lower bound for constant weighted -codes.

###### Theorem III.4.

For all positive integers and , there exists a constant , such that when , we have

 Mw(m,d,x)≥(1−o(1))(m⌈w/(x+d−1)⌉)(w⌈w/(x+d−1)⌉). (8)
###### Proof of Theorem iii.4.

For given and , we only need to show that the indicator vectors of a -packing of size form an -code of constant weight .

Consider a -packing of size , fix any distinct -subsets from . For each , denote as its indicator vector. Assume that there exist distinct -subsets in for some , such that

 (x⋁i=1vPi)∨(l⨁j=1vQj)=x⋁i=1vPi.

Then, denote as the subset of with indicator vector , we have

 (Q0∩Qj)⊆x⋃i=1(Pi∩Qj),

for every . W.l.o.g, take . Since , we have

 |Q1∖Q0|≤|l⋃j=2(Q1∩Qj)|.

Therefore,

 |Q1|=|Q1∩Q0|+|Q1∖Q0| ≤|x⋃i=1(Q1∩Pi)|+|l⋃j=2(Q1∩Qj)| ≤x∑i=1|(Q1∩Pi)|+l∑j=2|(Q1∩Qi)| ≤(x+l−1)⋅(⌈wx+d−1⌉−1)

this leads to a contradiction, which indicates that is an -code of constant weight . This completes the proof. ∎

### Iii-C Explicit constructions of constant weighted X-codes

#### Iii-C1 Construction of constant weighted X-codes with d=3 and x=2

In this part, based on some results from additive combinatorics, we shall prove the following asymptotic lower bound for constant weighted -codes with and .

###### Theorem III.5.

For any and , there exists a constant , such that for , there exists an -code of constant weight .

For the proof of Theorem III.5, we need the following lemma from [11].

###### Lemma III.6.

[11] For positive integers and , there exists a set of positive integers of size

 |A|≥meclogw√logm

for some absolute constant , such that contains no three terms of any arithmetic progressions of length .

The specific construction of the set from Lemma III.6 can be regarded as an extension of the -AP-free subset of given by Behrend [4]: Let and , for any , can be written as

 a=a0+a1(2βw)+a2(2βw)2+⋯+ak(2βw)k.

Set , where . For , set

 A=Am,β,s={a: 1≤a≤n, 0≤ai≤d for all i, (N(α))2=s}.

Based on the construction from Lemma III.6, we proceed to prove Theorem III.5.

###### Proof of Theorem iii.5.

Let and be the subset constructed from Lemma III.6 such that contains no three terms of any progressions of length .

Take distinct sets , where . Define

 P={(x,x+a,…,x+(w−1)a):1≤x≤m′,a∈A},

as a family of ordered -subsets in , then . Since for some , we have for every and .

Now, considering the indicator vectors corresponding to the -subsets in , we have the following claim.

Claim 1. is an -code of constant weight .

For each , denote as its indicator vector. First, noted that for any two distinct and , thus the superimposed sum of any two vectors in can not cover any other vector in .

Meanwhile, we can obtain . Therefore, when , we have . If there exist four distinct such that is covered by , one of the four intersections , , , must be strictly larger than one, which leads to a contradiction. When , if there exist distinct such that is covered by , then we have , , , , , where for , elements are pairwise distinct for some different . From the definition of , we have

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩θ0=x1+i0a1=x2+i0a2;θ1=x1+i1a1=x3+i1a3;θ2=x1+i2a1=x4+i2a4;θ3=x2+i3a2=x3+i3a3;θ4=x2+i4a2=x4+i4a4,

where element is the leading term in , and element is the common difference corresponding to . Combining the first two identities with the one above, we have

 (i1−i0)a1+(i0−i3)a2=(i1−i3)a3.

This means that are three distinct terms in a -AP, which contradicts the choice of set .

For any three distinct , we have . Therefore, when , the addition of any three distinct vectors in can not be covered by the superimposed sum of any two other vectors.

Now, assume that there exist such that is covered by .

When , we have either or . For the case , we have , , , for three pairwise distinct . For the case , we have , , , , , for five pairwise distinct . For both cases, we have three distinct s pairwise intersecting at three distinct elements s. From the analysis above, we know that this will lead to contains three distinct terms in a -AP, a contradiction.

When , we have either or . For the case , we have for some and , a contradiction. For the case , we have , , and for each , where elements are pairwise distinct. From the definition of , we have

 ⎧⎨⎩θ12=x1+s12a1=x2+s12a2, for some%  0≤s12≤w−1;θ14=x1+s14a1=x4+s14a4, for some 0≤s14≤w−1;θ24=x2+s24a2=x4+s24a4, for some 0≤s24≤w−1.

And this also leads to contains three distinct terms in a -AP, a contradiction.

For the cases when and , we have . Therefore, for both cases, there exist such that for some and , a contradiction.

In conclusion, any three or fewer vectors in can not be covered by the superimposed sum of any other two vectors. Therefore, Claim 1 is verified and this completes the proof. ∎

###### Remark III.7.

According to the upper bound given by (6), we have

 ⎧⎨⎩M3(m,3,2)≤m(m−1)6,M4(m,3,2)≤(m−1)(m−2)6.

This implies that for cases and , the lower bounds from Theorem III.5 are nearly optimal. For cases when , Theorem III.4 provide better lower bounds , but the gaps between the upper bounds and the lower bounds are still quite large.

It is also worth noting that, the construction from Theorem III.5 was originally proposed by Erdös et al. [11] to construct -uniform hypergraphs on vertices such that no vertices span or more hyperedges. This kind of hypergraphs is a special kind of sparse hypergraphs which we will discuss later in Section III.D.

#### Iii-C2 Construction of constant weighted X-codes with d=5 and x=2

Before we present the construction, we shall prove a proposition which establishes a connection between constant weighted -codes with and uniform hypergraphs of girth five.

Given a -uniform hypergraph and a positive integer , a cycle of length in , denoted by , is an alternating sequence of distinct vertices and hyperedges of the form: , such that for each and . A linear path of length , denoted by , is an alternating sequence of distinct vertices and hyperedges of the form: , such that for each and whenever . And the girth of hypergraph is the minimum length of a cycle in .

###### Proposition III.8.

Let be a positive integer. For any -uniform hypergraph of girth at least , the set of all the indicator vectors of hyperedges in forms a -code of constant weight .

###### Proof of Proposition iii.8.

First, noted that the girth of is at least , we know that is a linear hypergraph, i.e., for any . Hence, if we denote as the the indicator vector of hyperedge , then for any , we have

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩W(⨁i∈I2vEi)≥2(w−1), % for any 2-subset I2⊆[5];W(⨁i∈I3vEi)≥3(w−2), for any 3-% subset I3⊆[5];W(⨁i∈I4vEi)≥4(w−3), for any 4-% subset I4⊆[5];W(⨁i∈I5vEi)≥5(w−4), for any 5-% subset I5⊆[5].

Case 1. Assume that there exist such that is covered by . From the linearity, can cover at most distinct vertices in each (), thus we have , which indicates that .

When the length of the longest linear path in the configuration formed by is at most , the assumption that is covered by forces (or ) together with three distinct hyperedges from to form a cycle of length , this contradicts the requirement that is a hypergraph of girth at least .

When the length of the longest linear path in the configuration formed by equals to , one can easily determine all the possible types of this configuration. As those shown in Fig. 1, there are four types. Based on this characterization, in order to cover all the vertices with odd degree, (or ) has to form a cycle of length at most with two or three distinct hyperedges from , a contradiction.

Similarly, when the length of the longest linear path in the configuration formed by is , then this configuration itself is a linear path of length , as shown in Fig. 2. Again, the assumption forces the existence of a cycle of length at most , a contradiction.

Case 2. Assume that there exist such that is covered by . From the linearity, we have , which indicates that or . Similar to the analysis in Case 1, for all these three cases, either form a cycle of length four in