Best Known (57−9, 57, s)-Nets in Base 3
(57−9, 57, 4926)-Net over F3 — Constructive and digital
Digital (48, 57, 4926)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 4926, F3, 9, 9) (dual of [(4926, 9), 44277, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(357, 19705, F3, 9) (dual of [19705, 19648, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(356, 19704, F3, 9) (dual of [19704, 19648, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(355, 19684, F3, 9) (dual of [19684, 19629, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(356, 19704, F3, 9) (dual of [19704, 19648, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(357, 19705, F3, 9) (dual of [19705, 19648, 10]-code), using
(57−9, 57, 11082)-Net over F3 — Digital
Digital (48, 57, 11082)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(357, 11082, F3, 9) (dual of [11082, 11025, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 19705, F3, 9) (dual of [19705, 19648, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(356, 19704, F3, 9) (dual of [19704, 19648, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(355, 19684, F3, 9) (dual of [19684, 19629, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(356, 19704, F3, 9) (dual of [19704, 19648, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 19705, F3, 9) (dual of [19705, 19648, 10]-code), using
(57−9, 57, 5293221)-Net in Base 3 — Upper bound on s
There is no (48, 57, 5293222)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 56, 5293222)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 523 347706 275870 513453 988617 > 356 [i]