Best Known (170−28, 170, s)-Nets in Base 8
(170−28, 170, 149797)-Net over F8 — Constructive and digital
Digital (142, 170, 149797)-net over F8, using
- 81 times duplication [i] based on digital (141, 169, 149797)-net over F8, using
- net defined by OOA [i] based on linear OOA(8169, 149797, F8, 28, 28) (dual of [(149797, 28), 4194147, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8169, 2097158, F8, 28) (dual of [2097158, 2096989, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 2097159, F8, 28) (dual of [2097159, 2096990, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(8169, 2097152, F8, 28) (dual of [2097152, 2096983, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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(80, 7, F8, 0) (dual of [7, 7, 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(27) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8169, 2097159, F8, 28) (dual of [2097159, 2096990, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(8169, 2097158, F8, 28) (dual of [2097158, 2096989, 29]-code), using
- net defined by OOA [i] based on linear OOA(8169, 149797, F8, 28, 28) (dual of [(149797, 28), 4194147, 29]-NRT-code), using
(170−28, 170, 1117568)-Net over F8 — Digital
Digital (142, 170, 1117568)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 1117568, F8, 28) (dual of [1117568, 1117398, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 2097167, F8, 28) (dual of [2097167, 2096997, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(8169, 2097152, F8, 28) (dual of [2097152, 2096983, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 2097167, F8, 28) (dual of [2097167, 2096997, 29]-code), using
(170−28, 170, large)-Net in Base 8 — Upper bound on s
There is no (142, 170, large)-net in base 8, because
- 26 times m-reduction [i] would yield (142, 144, large)-net in base 8, but