Best Known (82, 94, s)-Nets in Base 3
(82, 94, 29529)-Net over F3 — Constructive and digital
Digital (82, 94, 29529)-net over F3, using
- 1 times m-reduction [i] based on digital (82, 95, 29529)-net over F3, using
- net defined by OOA [i] based on linear OOA(395, 29529, F3, 13, 13) (dual of [(29529, 13), 383782, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(395, 177175, F3, 13) (dual of [177175, 177080, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 177176, F3, 13) (dual of [177176, 177081, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(367, 177148, F3, 9) (dual of [177148, 177081, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(395, 177176, F3, 13) (dual of [177176, 177081, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(395, 177175, F3, 13) (dual of [177175, 177080, 14]-code), using
- net defined by OOA [i] based on linear OOA(395, 29529, F3, 13, 13) (dual of [(29529, 13), 383782, 14]-NRT-code), using
(82, 94, 88588)-Net over F3 — Digital
Digital (82, 94, 88588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(394, 88588, F3, 2, 12) (dual of [(88588, 2), 177082, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(394, 177176, F3, 12) (dual of [177176, 177082, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(394, 177177, F3, 12) (dual of [177177, 177083, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(390, 177171, F3, 12) (dual of [177171, 177081, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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 Ce(12) ⊂ Ce(9) [i] based on
- linear OA(390, 177173, F3, 9) (dual of [177173, 177083, 10]-code), using Gilbert–Varšamov bound and bm = 390 > Vbs−1(k−1) = 6163 413608 145158 977671 819957 801215 511441 [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(390, 177171, F3, 12) (dual of [177171, 177081, 13]-code), using
- construction X with Varšamov bound [i] based on
- discarding factors / shortening the dual code based on linear OA(394, 177177, F3, 12) (dual of [177177, 177083, 13]-code), using
- OOA 2-folding [i] based on linear OA(394, 177176, F3, 12) (dual of [177176, 177082, 13]-code), using
(82, 94, large)-Net in Base 3 — Upper bound on s
There is no (82, 94, large)-net in base 3, because
- 10 times m-reduction [i] would yield (82, 84, large)-net in base 3, but