Best Known (124−27, 124, s)-Nets in Base 9
(124−27, 124, 4543)-Net over F9 — Constructive and digital
Digital (97, 124, 4543)-net over F9, using
- 92 times duplication [i] based on digital (95, 122, 4543)-net over F9, using
- net defined by OOA [i] based on linear OOA(9122, 4543, F9, 27, 27) (dual of [(4543, 27), 122539, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9122, 59060, F9, 27) (dual of [59060, 58938, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9122, 59061, F9, 27) (dual of [59061, 58939, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- 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(9111, 59050, F9, 25) (dual of [59050, 58939, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9122, 59061, F9, 27) (dual of [59061, 58939, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9122, 59060, F9, 27) (dual of [59060, 58938, 28]-code), using
- net defined by OOA [i] based on linear OOA(9122, 4543, F9, 27, 27) (dual of [(4543, 27), 122539, 28]-NRT-code), using
(124−27, 124, 59067)-Net over F9 — Digital
Digital (97, 124, 59067)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9124, 59067, F9, 27) (dual of [59067, 58943, 28]-code), using
- 1 times truncation [i] based on linear OA(9125, 59068, F9, 28) (dual of [59068, 58943, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- linear OA(9121, 59049, F9, 28) (dual of [59049, 58928, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(27) ⊂ Ce(23) [i] based on
- 1 times truncation [i] based on linear OA(9125, 59068, F9, 28) (dual of [59068, 58943, 29]-code), using
(124−27, 124, large)-Net in Base 9 — Upper bound on s
There is no (97, 124, large)-net in base 9, because
- 25 times m-reduction [i] would yield (97, 99, large)-net in base 9, but