Best Known (98, 98+15, s)-Nets in Base 3
(98, 98+15, 25310)-Net over F3 — Constructive and digital
Digital (98, 113, 25310)-net over F3, using
- 31 times duplication [i] based on digital (97, 112, 25310)-net over F3, using
- net defined by OOA [i] based on linear OOA(3112, 25310, F3, 15, 15) (dual of [(25310, 15), 379538, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3112, 177171, F3, 15) (dual of [177171, 177059, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3111, 177148, F3, 15) (dual of [177148, 177037, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(389, 177148, F3, 13) (dual of [177148, 177059, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(31, 23, F3, 1) (dual of [23, 22, 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 X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3112, 177171, F3, 15) (dual of [177171, 177059, 16]-code), using
- net defined by OOA [i] based on linear OOA(3112, 25310, F3, 15, 15) (dual of [(25310, 15), 379538, 16]-NRT-code), using
(98, 98+15, 68491)-Net over F3 — Digital
Digital (98, 113, 68491)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3113, 68491, F3, 2, 15) (dual of [(68491, 2), 136869, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3113, 88586, F3, 2, 15) (dual of [(88586, 2), 177059, 16]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3112, 88586, F3, 2, 15) (dual of [(88586, 2), 177060, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3112, 177172, F3, 15) (dual of [177172, 177060, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3111, 177148, F3, 15) (dual of [177148, 177037, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(389, 177148, F3, 13) (dual of [177148, 177059, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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
- OOA 2-folding [i] based on linear OA(3112, 177172, F3, 15) (dual of [177172, 177060, 16]-code), using
- 31 times duplication [i] based on linear OOA(3112, 88586, F3, 2, 15) (dual of [(88586, 2), 177060, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3113, 88586, F3, 2, 15) (dual of [(88586, 2), 177059, 16]-NRT-code), using
(98, 98+15, large)-Net in Base 3 — Upper bound on s
There is no (98, 113, large)-net in base 3, because
- 13 times m-reduction [i] would yield (98, 100, large)-net in base 3, but