Best Known (119, 119+27, s)-Nets in Base 8
(119, 119+27, 20167)-Net over F8 — Constructive and digital
Digital (119, 146, 20167)-net over F8, using
- 81 times duplication [i] based on digital (118, 145, 20167)-net over F8, using
- net defined by OOA [i] based on linear OOA(8145, 20167, F8, 27, 27) (dual of [(20167, 27), 544364, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8145, 262172, F8, 27) (dual of [262172, 262027, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 262174, F8, 27) (dual of [262174, 262029, 28]-code), using
- construction X applied to Ce(26) ⊂ 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(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(86, 30, F8, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8145, 262174, F8, 27) (dual of [262174, 262029, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8145, 262172, F8, 27) (dual of [262172, 262027, 28]-code), using
- net defined by OOA [i] based on linear OOA(8145, 20167, F8, 27, 27) (dual of [(20167, 27), 544364, 28]-NRT-code), using
(119, 119+27, 251434)-Net over F8 — Digital
Digital (119, 146, 251434)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8146, 251434, F8, 27) (dual of [251434, 251288, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8146, 262158, F8, 27) (dual of [262158, 262012, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(8145, 262145, F8, 27) (dual of [262145, 262000, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8133, 262145, F8, 25) (dual of [262145, 262012, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 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(8146, 262158, F8, 27) (dual of [262158, 262012, 28]-code), using
(119, 119+27, large)-Net in Base 8 — Upper bound on s
There is no (119, 146, large)-net in base 8, because
- 25 times m-reduction [i] would yield (119, 121, large)-net in base 8, but