## Construction X4 with Extended Narrow-Sense BCH Codes

Let C_{1}ʹ denote an [*s*_{1}, *n*_{1}, *d*_{1}ʹ]-code which is a subcode of an [*s*_{1}, *n*_{1} + *Δn*, *d*_{1}]-code C_{1}. Similarly, let C_{2}ʹ denote an [*s*_{2}, *n*_{2}, *d*_{2}ʹ]-code which is a subcode of an [*s*_{2}, *n*_{2} + *Δn*, *d*_{2}]-code C_{2}. Then construction X4 allows the construction of an [*s*_{1} + *s*_{2}, *n*_{1} + *n*_{2} + *Δn*, *d*]-code C with

*d*= min{

*d*

_{1}ʹ,

*d*

_{2}ʹ,

*d*

_{1}+

*d*

_{2}},

where a value of *d*_{i}ʹ = *s*_{i} + 1 (i.e., C_{i}ʹ = { 0}) can be treated as *d*_{i}ʹ = ∞ [1].

Correspondingly, if A_{1} denotes a linear orthogonal array OA(*b*^{m1}, *s*_{1},**F**_{b}, *k*_{1}) which is a subspace of a linear OA(*b*^{m1+Δm}, *s*_{1},**F**_{b}, *k*_{1}ʹ) A_{1}ʹ, and if A_{2} denotes a linear OA(*b*^{m2}, *s*_{2},**F**_{b}, *k*_{2}) which is a subspace of a linear OA(*b*^{m2+Δm}, *s*_{2},**F**_{b}, *k*_{2}ʹ) A_{2}ʹ, then a linear OA(*b*^{m1+m2+Δm}, *s*_{1} + *s*_{2},**F**_{b}, *k*) with

*k*= min{

*k*

_{1}ʹ,

*k*

_{2}ʹ,

*k*

_{1}+

*k*

_{2}+1}

can be constructed, where a value of *k*_{i}ʹ = *s*_{i} (i.e., A_{i}ʹ = **F**_{b}^{si}) can be treated as *k*_{i}ʹ = ∞.

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* ∈ {1, 2} let **G**_{i}ʹ denote generator matrices of C_{i}ʹ and let **G**_{i} denote generator matrices for C_{i} such that the first *n*_{i} rows of **G**_{i} are equal to **G**_{i}ʹ. Furthermore let **G**_{i}” denote the remaining *Δn* rows of **G**_{i}. 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 C_{1}, C_{1}ʹ, C_{2}, C_{2}ʹ instead of the generator matrices of C_{1}ʹ, C_{1}, C_{2}ʹ, C_{2}, 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 C_{i}ʹ and C_{i} are (*s*_{i}, *N*_{i}, *d*_{i}ʹ)- and (*s*_{i}, *N*_{i}⋅*ΔN*, *d*_{i})-codes, respectively, and C_{i} is the union of *ΔN* disjoint translates of C_{i}ʹ, i.e,

_{i}= (

**v**_{j}

^{(i)}+ C

_{i}ʹ),

then

**v**_{j}

^{(1)}+

**x**_{1},

**v**_{j}

^{(2)}+

**x**_{2})

is an (*s*_{1} + *s*_{2}, *N*_{1}*N*_{1}⋅*ΔN*, *d*)-code.

### Construction for Non-Linear Orthogonal Arrays

Suppose that the A_{i} and A_{i}ʹ are OA(*M*_{i}, *s*_{i}, *S*_{b}, *k*_{i}) and OA(*M*_{i}⋅*ΔM*, *s*_{i}, *S*_{b}, *k*_{i}ʹ), respectively, and A_{i}ʹ is the union of *ΔM* disjoint translates of A_{i}, i.e.,

_{i}ʹ = (

**v**_{j}

^{(i)}+ A

_{i}),

then

**v**_{j}

^{(1)}+

**x**_{1},

**v**_{j}

^{(2)}+

**x**_{2})

is an OA(*M*_{1}*M*_{2}⋅*ΔM*, *s*_{1} + *s*_{2}, *S*_{b}, *k*) [2, Theorem 10.5].

### Special Cases

If C

_{i}ʹ = { 0} (or A_{i}ʹ =**F**_{q}^{si}) for one*i*, construction X applied to the remaining three codes / OAs follows.If C

_{i}ʹ = { 0} (or A_{i}ʹ =**F**_{q}^{si}) for*i*= 1 and*i*= 2, juxtaposition applied to C_{1}and C_{2}(or A_{1}and A_{2}) follows.If C

_{1}= C_{1}ʹ and C_{2}= C_{2}ʹ, the direct product of C_{1}and C_{2}follows.

### Applications

Construction X4 is not used in its general form by MinT. In addition to the cases covered by Construction X, it is applied to all pairs C_{1}ʹ ⊂ C_{1} (or A_{1} ⊂ A_{1}ʹ) considered by construction X, but only for the following pairs of codes C_{2}ʹ ⊂ C_{2} (or their dual OAs A_{2} ⊂ A_{2}ʹ):

If

*b*≠ 2 or*s*_{2}is even, then the [*s*_{2},*s*_{2}− 1, 2]-parity-check code C_{2}contains an [*s*_{2}, 1,*s*_{2}]-code C_{2}ʹ.For

*b*= 2 and*s*_{2}odd, the [*s*_{2},*s*_{2}− 1, 2]-parity-check code C_{2}contains only an [*s*_{2}, 1,*s*_{2}− 1]-code C_{2}ʹ.Truncating the ternary Hamming code H(3, 3) yields a [12, 9, 3]-code C

_{2}containing a [12, 1, 12]-code C_{2}ʹ .

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

### References

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Construction X4 with Extended Narrow-Sense BCH Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CConsX4-NS.html