Best Known (170−27, 170, s)-Nets in Base 8
(170−27, 170, 161322)-Net over F8 — Constructive and digital
Digital (143, 170, 161322)-net over F8, using
- 81 times duplication [i] based on digital (142, 169, 161322)-net over F8, using
- net defined by OOA [i] based on linear OOA(8169, 161322, F8, 27, 27) (dual of [(161322, 27), 4355525, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8169, 2097187, F8, 27) (dual of [2097187, 2097018, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 2097194, F8, 27) (dual of [2097194, 2097025, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [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(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 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(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8169, 2097194, F8, 27) (dual of [2097194, 2097025, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8169, 2097187, F8, 27) (dual of [2097187, 2097018, 28]-code), using
- net defined by OOA [i] based on linear OOA(8169, 161322, F8, 27, 27) (dual of [(161322, 27), 4355525, 28]-NRT-code), using
(170−27, 170, 1851026)-Net over F8 — Digital
Digital (143, 170, 1851026)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 1851026, F8, 27) (dual of [1851026, 1850856, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 2097168, F8, 27) (dual of [2097168, 2096998, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(8169, 2097153, F8, 27) (dual of [2097153, 2096984, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8155, 2097153, F8, 25) (dual of [2097153, 2096998, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 2097168, F8, 27) (dual of [2097168, 2096998, 28]-code), using
(170−27, 170, large)-Net in Base 8 — Upper bound on s
There is no (143, 170, large)-net in base 8, because
- 25 times m-reduction [i] would yield (143, 145, large)-net in base 8, but