## ConstructionÂ X4 with Extended Narrow-Sense BCH Codes

Let C1Ê¹ denote an [s1, n1, d1Ê¹]-code which is a subcode of an [s1, n1 + Î”n, d1]-code C1. Similarly, let C2Ê¹ denote an [s2, n2, d2Ê¹]-code which is a subcode of an [s2, n2 + Î”n, d2]-code C2. Then constructionÂ X4 allows the construction of an [s1 + s2, n1 + n2 + Î”n, d]-code C with

d = min{d1Ê¹, d2Ê¹, d1 + d2},

where a value of diÊ¹ = si + 1 (i.e., CiÊ¹ = { 0}) can be treated as diÊ¹ = âˆž [1].

Correspondingly, if A1 denotes a linear orthogonal array OA(bm1, s1,Fb, k1) which is a subspace of a linear OA(bm1+Î”m, s1,Fb, k1Ê¹) A1Ê¹, and if A2 denotes a linear OA(bm2, s2,Fb, k2) which is a subspace of a linear OA(bm2+Î”m, s2,Fb, k2Ê¹) A2Ê¹, then a linear OA(bm1+m2+Î”m, s1 + s2,Fb, k) with

k = min{k1Ê¹, k2Ê¹, k1 + k2 +1}

can be constructed, where a value of kiÊ¹ = si (i.e., AiÊ¹ = Fbsi) can be treated as kiÊ¹ = âˆž.

Equivalent results hold also for non-linear codes as well as for non-linear OAs, provided that the larger codes/OAs can be partitioned in translates of the smaller ones (see below).

### Construction

For i âˆˆ {1, 2} let GiÊ¹ denote generator matrices of CiÊ¹ and let Gi denote generator matrices for Ci such that the first ni rows of Gi are equal to GiÊ¹. Furthermore let Giâ€ denote the remaining Î”n rows of Gi. Then a generator matrix of C is given by

G = .

The construction is self-dual, i.e., if the same construction is performed using the check-matrices of C1, C1Ê¹, C2, C2Ê¹ instead of the generator matrices of C1Ê¹, C1, C2Ê¹, C2, respectively, a check-matrix of C is obtained. In particular, this implies that this construction can be applied directly to the generator matrices of orthogonal arrays.

### Construction for Non-Linear Codes

Suppose that the codes CiÊ¹ and Ci are (si, Ni, diÊ¹)- and (si, Niâ‹…Î”N, di)-codes, respectively, and Ci is the union of Î”N disjoint translates of CiÊ¹, i.e,

Ci = (vj(i) + CiÊ¹),

then

C := (vj(1) + x1,vj(2) + x2)

is an (s1 + s2, N1N1â‹…Î”N, d)-code.

### Construction for Non-Linear Orthogonal Arrays

Suppose that the Ai and AiÊ¹ are OA(Mi, si, Sb, ki) and OA(Miâ‹…Î”M, si, Sb, kiÊ¹), respectively, and AiÊ¹ is the union of Î”M disjoint translates of Ai, i.e.,

AiÊ¹ = (vj(i) + Ai),

then

A := (vj(1) + x1,vj(2) + x2)

is an OA(M1M2â‹…Î”M, s1 + s2, Sb, k) [2, TheoremÂ 10.5].

### Special Cases

• If CiÊ¹ = { 0} (or AiÊ¹ = Fqsi) for one i, constructionÂ X applied to the remaining three codes / OAs follows.

• If CiÊ¹ = { 0} (or AiÊ¹ = Fqsi) for i = 1 and i = 2, juxtaposition applied to C1 and C2 (or A1 and A2) follows.

• If C1 = C1Ê¹ and C2 = C2Ê¹, the direct product of C1 and C2 follows.

### Applications

ConstructionÂ X4 is not used in its general form by MinT. In addition to the cases covered by ConstructionÂ X, it is applied to all pairs C1Ê¹ âŠ‚ C1 (or A1 âŠ‚ A1Ê¹) considered by constructionÂ X, but only for the following pairs of codes C2Ê¹ âŠ‚ C2 (or their dual OAs A2 âŠ‚ A2Ê¹):

• If b â‰  2 or s2 is even, then the [s2, s2 âˆ’ 1, 2]-parity-check code C2 contains an [s2, 1, s2]-code C2Ê¹.

• For b = 2 and s2 odd, the [s2, s2 âˆ’ 1, 2]-parity-check code C2 contains only an [s2, 1, s2 âˆ’ 1]-code C2Ê¹.

• Truncating the ternary Hamming code H(3, 3) yields a [12, 9, 3]-code C2 containing a [12, 1, 12]-code C2Ê¹ .

These are exactly the cases considered in [3, SectionÂ 3.6].

### References

 [1] Neil J.Â A. Sloane, S.Â M. Reddy, and Chin-Long Chen.New binary codes.IEEE Transactions on Information Theory, 18(4):503â€“510, July 1972. [2] A.Â S. Hedayat, Neil J.Â A. Sloane, and John Stufken.Orthogonal Arrays.Springer Series in Statistics. Springer-Verlag, 1999. [3] Yves Edel.Eine Verallgemeinerung von BCH-Codes.PhD thesis, Ruprecht-Karls-UniversitÃ¤t, Heidelberg, 1996.