Best Known (90, 105, s)-Nets in Base 3
(90, 105, 8439)-Net over F3 — Constructive and digital
Digital (90, 105, 8439)-net over F3, using
- 31 times duplication [i] based on digital (89, 104, 8439)-net over F3, using
- net defined by OOA [i] based on linear OOA(3104, 8439, F3, 15, 15) (dual of [(8439, 15), 126481, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3104, 59074, F3, 15) (dual of [59074, 58970, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3101, 59050, F3, 15) (dual of [59050, 58949, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3104, 59074, F3, 15) (dual of [59074, 58970, 16]-code), using
- net defined by OOA [i] based on linear OOA(3104, 8439, F3, 15, 15) (dual of [(8439, 15), 126481, 16]-NRT-code), using
(90, 105, 29538)-Net over F3 — Digital
Digital (90, 105, 29538)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3105, 29538, F3, 2, 15) (dual of [(29538, 2), 58971, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3105, 59076, F3, 15) (dual of [59076, 58971, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3101, 59050, F3, 15) (dual of [59050, 58949, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3102, 59073, F3, 12) (dual of [59073, 58971, 13]-code), using Gilbert–Varšamov bound and bm = 3102 > Vbs−1(k−1) = 1 566782 536444 133782 911502 484377 699469 910493 064961 [i]
- linear OA(32, 3, F3, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,3)), using
- dual of repetition code with length 3 [i]
- Reed–Solomon code RS(1,3) [i]
- linear OA(3102, 59072, F3, 15) (dual of [59072, 58970, 16]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(3105, 59076, F3, 15) (dual of [59076, 58971, 16]-code), using
(90, 105, large)-Net in Base 3 — Upper bound on s
There is no (90, 105, large)-net in base 3, because
- 13 times m-reduction [i] would yield (90, 92, large)-net in base 3, but