Best Known (81, 81+15, s)-Nets in Base 3
(81, 81+15, 2815)-Net over F3 — Constructive and digital
Digital (81, 96, 2815)-net over F3, using
- 32 times duplication [i] based on digital (79, 94, 2815)-net over F3, using
- net defined by OOA [i] based on linear OOA(394, 2815, F3, 15, 15) (dual of [(2815, 15), 42131, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(394, 19706, F3, 15) (dual of [19706, 19612, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(391, 19684, F3, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(394, 19706, F3, 15) (dual of [19706, 19612, 16]-code), using
- net defined by OOA [i] based on linear OOA(394, 2815, F3, 15, 15) (dual of [(2815, 15), 42131, 16]-NRT-code), using
(81, 81+15, 9855)-Net over F3 — Digital
Digital (81, 96, 9855)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(396, 9855, F3, 2, 15) (dual of [(9855, 2), 19614, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(396, 19710, F3, 15) (dual of [19710, 19614, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(391, 19684, F3, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(392, 19706, F3, 12) (dual of [19706, 19614, 13]-code), using Gilbert–Varšamov bound and bm = 392 > Vbs−1(k−1) = 8 900695 728292 029200 450900 974131 289283 431203 [i]
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- linear OA(392, 19704, F3, 15) (dual of [19704, 19612, 16]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(396, 19710, F3, 15) (dual of [19710, 19614, 16]-code), using
(81, 81+15, 5047834)-Net in Base 3 — Upper bound on s
There is no (81, 96, 5047835)-net in base 3, because
- 1 times m-reduction [i] would yield (81, 95, 5047835)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2120 896140 334428 884141 103937 827203 328913 965795 > 395 [i]