Best Known (117−21, 117, s)-Nets in Base 8
(117−21, 117, 26218)-Net over F8 — Constructive and digital
Digital (96, 117, 26218)-net over F8, using
- 81 times duplication [i] based on digital (95, 116, 26218)-net over F8, using
- net defined by OOA [i] based on linear OOA(8116, 26218, F8, 21, 21) (dual of [(26218, 21), 550462, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8116, 262181, F8, 21) (dual of [262181, 262065, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(8109, 262144, F8, 21) (dual of [262144, 262035, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(8116, 262181, F8, 21) (dual of [262181, 262065, 22]-code), using
- net defined by OOA [i] based on linear OOA(8116, 26218, F8, 21, 21) (dual of [(26218, 21), 550462, 22]-NRT-code), using
(117−21, 117, 262183)-Net over F8 — Digital
Digital (96, 117, 262183)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 262183, F8, 21) (dual of [262183, 262066, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8116, 262181, F8, 21) (dual of [262181, 262065, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(8109, 262144, F8, 21) (dual of [262144, 262035, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(8116, 262182, F8, 20) (dual of [262182, 262066, 21]-code), using Gilbert–Varšamov bound and bm = 8116 > Vbs−1(k−1) = 841225 577927 399353 496156 463638 310786 893012 991560 046145 182004 640653 015003 040084 676041 215473 358640 122424 [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(8116, 262181, F8, 21) (dual of [262181, 262065, 22]-code), using
- construction X with Varšamov bound [i] based on
(117−21, 117, large)-Net in Base 8 — Upper bound on s
There is no (96, 117, large)-net in base 8, because
- 19 times m-reduction [i] would yield (96, 98, large)-net in base 8, but