Best Known (140, 140+27, s)-Nets in Base 8
(140, 140+27, 161321)-Net over F8 — Constructive and digital
Digital (140, 167, 161321)-net over F8, using
- 81 times duplication [i] based on digital (139, 166, 161321)-net over F8, using
- net defined by OOA [i] based on linear OOA(8166, 161321, F8, 27, 27) (dual of [(161321, 27), 4355501, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8166, 2097174, F8, 27) (dual of [2097174, 2097008, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8166, 2097177, F8, 27) (dual of [2097177, 2097011, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(84, 25, F8, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,8)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8166, 2097177, F8, 27) (dual of [2097177, 2097011, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8166, 2097174, F8, 27) (dual of [2097174, 2097008, 28]-code), using
- net defined by OOA [i] based on linear OOA(8166, 161321, F8, 27, 27) (dual of [(161321, 27), 4355501, 28]-NRT-code), using
(140, 140+27, 1442251)-Net over F8 — Digital
Digital (140, 167, 1442251)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8167, 1442251, F8, 27) (dual of [1442251, 1442084, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8167, 2097179, F8, 27) (dual of [2097179, 2097012, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8134, 2097152, F8, 22) (dual of [2097152, 2097018, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(84, 26, F8, 3) (dual of [26, 22, 4]-code or 26-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(8167, 2097179, F8, 27) (dual of [2097179, 2097012, 28]-code), using
(140, 140+27, large)-Net in Base 8 — Upper bound on s
There is no (140, 167, large)-net in base 8, because
- 25 times m-reduction [i] would yield (140, 142, large)-net in base 8, but