Best Known (119, 119+27, s)-Nets in Base 9
(119, 119+27, 40881)-Net over F9 — Constructive and digital
Digital (119, 146, 40881)-net over F9, using
- net defined by OOA [i] based on linear OOA(9146, 40881, F9, 27, 27) (dual of [(40881, 27), 1103641, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9146, 531454, F9, 27) (dual of [531454, 531308, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9146, 531455, F9, 27) (dual of [531455, 531309, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(9145, 531442, F9, 27) (dual of [531442, 531297, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(9133, 531442, F9, 25) (dual of [531442, 531309, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 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(9146, 531455, F9, 27) (dual of [531455, 531309, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9146, 531454, F9, 27) (dual of [531454, 531308, 28]-code), using
(119, 119+27, 435640)-Net over F9 — Digital
Digital (119, 146, 435640)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9146, 435640, F9, 27) (dual of [435640, 435494, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9146, 531455, F9, 27) (dual of [531455, 531309, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(9145, 531442, F9, 27) (dual of [531442, 531297, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(9133, 531442, F9, 25) (dual of [531442, 531309, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 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(9146, 531455, F9, 27) (dual of [531455, 531309, 28]-code), using
(119, 119+27, large)-Net in Base 9 — Upper bound on s
There is no (119, 146, large)-net in base 9, because
- 25 times m-reduction [i] would yield (119, 121, large)-net in base 9, but