Best Known (45, 45+11, s)-Nets in Base 8
(45, 45+11, 52431)-Net over F8 — Constructive and digital
Digital (45, 56, 52431)-net over F8, using
- net defined by OOA [i] based on linear OOA(856, 52431, F8, 11, 11) (dual of [(52431, 11), 576685, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(856, 262156, F8, 11) (dual of [262156, 262100, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(856, 262156, F8, 11) (dual of [262156, 262100, 12]-code), using
(45, 45+11, 195671)-Net over F8 — Digital
Digital (45, 56, 195671)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 195671, F8, 11) (dual of [195671, 195615, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
(45, 45+11, large)-Net in Base 8 — Upper bound on s
There is no (45, 56, large)-net in base 8, because
- 9 times m-reduction [i] would yield (45, 47, large)-net in base 8, but