Best Known (126−27, 126, s)-Nets in Base 9
(126−27, 126, 4544)-Net over F9 — Constructive and digital
Digital (99, 126, 4544)-net over F9, using
- 91 times duplication [i] based on digital (98, 125, 4544)-net over F9, using
- net defined by OOA [i] based on linear OOA(9125, 4544, F9, 27, 27) (dual of [(4544, 27), 122563, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9125, 59073, F9, 27) (dual of [59073, 58948, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9125, 59074, F9, 27) (dual of [59074, 58949, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [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(9101, 59050, F9, 23) (dual of [59050, 58949, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9125, 59074, F9, 27) (dual of [59074, 58949, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9125, 59073, F9, 27) (dual of [59073, 58948, 28]-code), using
- net defined by OOA [i] based on linear OOA(9125, 4544, F9, 27, 27) (dual of [(4544, 27), 122563, 28]-NRT-code), using
(126−27, 126, 59076)-Net over F9 — Digital
Digital (99, 126, 59076)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9126, 59076, F9, 27) (dual of [59076, 58950, 28]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9125, 59074, F9, 27) (dual of [59074, 58949, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [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(9101, 59050, F9, 23) (dual of [59050, 58949, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(9125, 59075, F9, 26) (dual of [59075, 58950, 27]-code), using Gilbert–Varšamov bound and bm = 9125 > Vbs−1(k−1) = 467 014785 933024 956013 050304 056713 981870 865971 129273 360904 924319 321551 973027 212429 544791 537570 944709 759758 676806 811217 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9125, 59074, F9, 27) (dual of [59074, 58949, 28]-code), using
- construction X with Varšamov bound [i] based on
(126−27, 126, large)-Net in Base 9 — Upper bound on s
There is no (99, 126, large)-net in base 9, because
- 25 times m-reduction [i] would yield (99, 101, large)-net in base 9, but