Best Known (132−29, 132, s)-Nets in Base 9
(132−29, 132, 4219)-Net over F9 — Constructive and digital
Digital (103, 132, 4219)-net over F9, using
- 92 times duplication [i] based on digital (101, 130, 4219)-net over F9, using
- net defined by OOA [i] based on linear OOA(9130, 4219, F9, 29, 29) (dual of [(4219, 29), 122221, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(9130, 59067, F9, 29) (dual of [59067, 58937, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9130, 59068, F9, 29) (dual of [59068, 58938, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(9126, 59049, F9, 29) (dual of [59049, 58923, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(94, 19, F9, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,9)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(9130, 59068, F9, 29) (dual of [59068, 58938, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(9130, 59067, F9, 29) (dual of [59067, 58937, 30]-code), using
- net defined by OOA [i] based on linear OOA(9130, 4219, F9, 29, 29) (dual of [(4219, 29), 122221, 30]-NRT-code), using
(132−29, 132, 58216)-Net over F9 — Digital
Digital (103, 132, 58216)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9132, 58216, F9, 29) (dual of [58216, 58084, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9132, 59061, F9, 29) (dual of [59061, 58929, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(9131, 59050, F9, 29) (dual of [59050, 58919, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(9121, 59050, F9, 27) (dual of [59050, 58929, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 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([0,14]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9132, 59061, F9, 29) (dual of [59061, 58929, 30]-code), using
(132−29, 132, large)-Net in Base 9 — Upper bound on s
There is no (103, 132, large)-net in base 9, because
- 27 times m-reduction [i] would yield (103, 105, large)-net in base 9, but