Best Known (113, 113+27, s)-Nets in Base 8
(113, 113+27, 20165)-Net over F8 — Constructive and digital
Digital (113, 140, 20165)-net over F8, using
- 81 times duplication [i] based on digital (112, 139, 20165)-net over F8, using
- net defined by OOA [i] based on linear OOA(8139, 20165, F8, 27, 27) (dual of [(20165, 27), 544316, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8139, 262146, F8, 27) (dual of [262146, 262007, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 262150, F8, 27) (dual of [262150, 262011, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(8139, 262144, F8, 27) (dual of [262144, 262005, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8139, 262150, F8, 27) (dual of [262150, 262011, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8139, 262146, F8, 27) (dual of [262146, 262007, 28]-code), using
- net defined by OOA [i] based on linear OOA(8139, 20165, F8, 27, 27) (dual of [(20165, 27), 544316, 28]-NRT-code), using
(113, 113+27, 152639)-Net over F8 — Digital
Digital (113, 140, 152639)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8140, 152639, F8, 27) (dual of [152639, 152499, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8140, 262157, F8, 27) (dual of [262157, 262017, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(8139, 262144, F8, 27) (dual of [262144, 262005, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8127, 262144, F8, 25) (dual of [262144, 262017, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(8140, 262157, F8, 27) (dual of [262157, 262017, 28]-code), using
(113, 113+27, large)-Net in Base 8 — Upper bound on s
There is no (113, 140, large)-net in base 8, because
- 25 times m-reduction [i] would yield (113, 115, large)-net in base 8, but