## ConstructionÂ X4 with Extended Narrow-Sense BCH Codes

Let C_{1}Ê¹ denote an [*s*_{1}, *n*_{1}, *d*_{1}Ê¹]-code which is a subcode of an [*s*_{1}, *n*_{1} + *Î”n*, *d*_{1}]-code C_{1}. Similarly, let C_{2}Ê¹ denote an [*s*_{2}, *n*_{2}, *d*_{2}Ê¹]-code which is a subcode of an [*s*_{2}, *n*_{2} + *Î”n*, *d*_{2}]-code C_{2}. Then constructionÂ X4 allows the construction of an [*s*_{1} + *s*_{2}, *n*_{1} + *n*_{2} + *Î”n*, *d*]-code C with

*d*= min{

*d*

_{1}Ê¹,

*d*

_{2}Ê¹,

*d*

_{1}+

*d*

_{2}},

where a value of *d*_{i}Ê¹ = *s*_{i} + 1 (i.e., C_{i}Ê¹ = { 0}) can be treated as *d*_{i}Ê¹ = âˆž [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+Î”m}, *s*_{1},**F**_{b}, *k*_{1}Ê¹) A_{1}Ê¹, 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+Î”m}, *s*_{2},**F**_{b}, *k*_{2}Ê¹) A_{2}Ê¹, then a linear OA(*b*^{m1+m2+Î”m}, *s*_{1} + *s*_{2},**F**_{b}, *k*) with

*k*= min{

*k*

_{1}Ê¹,

*k*

_{2}Ê¹,

*k*

_{1}+

*k*

_{2}+1}

can be constructed, where a value of *k*_{i}Ê¹ = *s*_{i} (i.e., A_{i}Ê¹ = **F**_{b}^{si}) can be treated as *k*_{i}Ê¹ = 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 **G**_{i}Ê¹ denote generator matrices of C_{i}Ê¹ 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}Ê¹. Furthermore let **G**_{i}â€ denote the remaining *Î”n* 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}Ê¹, C_{2}, C_{2}Ê¹ instead of the generator matrices of C_{1}Ê¹, C_{1}, C_{2}Ê¹, 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}Ê¹ and C_{i} are (*s*_{i}, *N*_{i}, *d*_{i}Ê¹)- and (*s*_{i}, *N*_{i}â‹…*Î”N*, *d*_{i})-codes, respectively, and C_{i} is the union of *Î”N* disjoint translates of C_{i}Ê¹, i.e,

_{i}= (

**v**_{j}

^{(i)}+ C

_{i}Ê¹),

then

**v**_{j}

^{(1)}+

**x**_{1},

**v**_{j}

^{(2)}+

**x**_{2})

is an (*s*_{1} + *s*_{2}, *N*_{1}*N*_{1}â‹…*Î”N*, *d*)-code.

### Construction for Non-Linear Orthogonal Arrays

Suppose that the A_{i} and A_{i}Ê¹ are OA(*M*_{i}, *s*_{i}, *S*_{b}, *k*_{i}) and OA(*M*_{i}â‹…*Î”M*, *s*_{i}, *S*_{b}, *k*_{i}Ê¹), respectively, and A_{i}Ê¹ is the union of *Î”M* disjoint translates of A_{i}, i.e.,

_{i}Ê¹ = (

**v**_{j}

^{(i)}+ A

_{i}),

then

**v**_{j}

^{(1)}+

**x**_{1},

**v**_{j}

^{(2)}+

**x**_{2})

is an OA(*M*_{1}*M*_{2}â‹…*Î”M*, *s*_{1} + *s*_{2}, *S*_{b}, *k*) [2, TheoremÂ 10.5].

### Special Cases

If C

_{i}Ê¹ = { 0} (or A_{i}Ê¹ =**F**_{q}^{si}) for one*i*, constructionÂ X applied to the remaining three codes / OAs follows.If C

_{i}Ê¹ = { 0} (or A_{i}Ê¹ =**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}Ê¹ and C_{2}= C_{2}Ê¹, the direct product of C_{1}and C_{2}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 C_{1}Ê¹ âŠ‚ C_{1} (or A_{1} âŠ‚ A_{1}Ê¹) considered by constructionÂ X, but only for the following pairs of codes C_{2}Ê¹ âŠ‚ C_{2} (or their dual OAs A_{2} âŠ‚ A_{2}Ê¹):

If

*b*â‰ 2 or*s*_{2}is even, then the [*s*_{2},*s*_{2}âˆ’ 1, 2]-parity-check code C_{2}contains an [*s*_{2}, 1,*s*_{2}]-code C_{2}Ê¹.For

*b*= 2 and*s*_{2}odd, the [*s*_{2},*s*_{2}âˆ’ 1, 2]-parity-check code C_{2}contains only an [*s*_{2}, 1,*s*_{2}âˆ’ 1]-code C_{2}Ê¹.Truncating the ternary Hamming code H(3, 3) yields a [12, 9, 3]-code C

_{2}containing a [12, 1, 12]-code C_{2}Ê¹ .

These are exactly the cases considered in [3, SectionÂ 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. “ConstructionÂ X4 with Extended Narrow-Sense BCH Codes.”
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Version: 2024-09-05.
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