Best Known (112, 112+23, s)-Nets in Base 8
(112, 112+23, 47663)-Net over F8 — Constructive and digital
Digital (112, 135, 47663)-net over F8, using
- 81 times duplication [i] based on digital (111, 134, 47663)-net over F8, using
- net defined by OOA [i] based on linear OOA(8134, 47663, F8, 23, 23) (dual of [(47663, 23), 1096115, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- net defined by OOA [i] based on linear OOA(8134, 47663, F8, 23, 23) (dual of [(47663, 23), 1096115, 24]-NRT-code), using
(112, 112+23, 524296)-Net over F8 — Digital
Digital (112, 135, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8135, 524296, F8, 23) (dual of [524296, 524161, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- linear OA(8134, 524295, F8, 22) (dual of [524295, 524161, 23]-code), using Gilbert–Varšamov bound and bm = 8134 > Vbs−1(k−1) = 14112 894707 310158 960625 521225 138116 752714 347085 375613 744584 737045 421774 379468 036074 399929 842346 330621 759362 065071 767552 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8134, 524294, F8, 23) (dual of [524294, 524160, 24]-code), using
- construction X with Varšamov bound [i] based on
(112, 112+23, large)-Net in Base 8 — Upper bound on s
There is no (112, 135, large)-net in base 8, because
- 21 times m-reduction [i] would yield (112, 114, large)-net in base 8, but