Best Known (117, 117+27, s)-Nets in Base 8
(117, 117+27, 20166)-Net over F8 — Constructive and digital
Digital (117, 144, 20166)-net over F8, using
- 82 times duplication [i] based on digital (115, 142, 20166)-net over F8, using
- net defined by OOA [i] based on linear OOA(8142, 20166, F8, 27, 27) (dual of [(20166, 27), 544340, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8142, 262159, F8, 27) (dual of [262159, 262017, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8142, 262160, F8, 27) (dual of [262160, 262018, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [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(8121, 262144, F8, 23) (dual of [262144, 262023, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 14, F8, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(81, 2, F8, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8142, 262160, F8, 27) (dual of [262160, 262018, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8142, 262159, F8, 27) (dual of [262159, 262017, 28]-code), using
- net defined by OOA [i] based on linear OOA(8142, 20166, F8, 27, 27) (dual of [(20166, 27), 544340, 28]-NRT-code), using
(117, 117+27, 212898)-Net over F8 — Digital
Digital (117, 144, 212898)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8144, 212898, F8, 27) (dual of [212898, 212754, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 262168, F8, 27) (dual of [262168, 262024, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [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(8121, 262144, F8, 23) (dual of [262144, 262023, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8115, 262144, F8, 22) (dual of [262144, 262029, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(84, 23, F8, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 262168, F8, 27) (dual of [262168, 262024, 28]-code), using
(117, 117+27, large)-Net in Base 8 — Upper bound on s
There is no (117, 144, large)-net in base 8, because
- 25 times m-reduction [i] would yield (117, 119, large)-net in base 8, but