Best Known (81−15, 81, s)-Nets in Base 3
(81−15, 81, 938)-Net over F3 — Constructive and digital
Digital (66, 81, 938)-net over F3, using
- net defined by OOA [i] based on linear OOA(381, 938, F3, 15, 15) (dual of [(938, 15), 13989, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(381, 6567, F3, 15) (dual of [6567, 6486, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 6569, F3, 15) (dual of [6569, 6488, 16]-code), using
- 1 times truncation [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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 Ce(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 6569, F3, 15) (dual of [6569, 6488, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(381, 6567, F3, 15) (dual of [6567, 6486, 16]-code), using
(81−15, 81, 3284)-Net over F3 — Digital
Digital (66, 81, 3284)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(381, 3284, F3, 2, 15) (dual of [(3284, 2), 6487, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(381, 6568, F3, 15) (dual of [6568, 6487, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 6569, F3, 15) (dual of [6569, 6488, 16]-code), using
- 1 times truncation [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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 Ce(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 6569, F3, 15) (dual of [6569, 6488, 16]-code), using
- OOA 2-folding [i] based on linear OA(381, 6568, F3, 15) (dual of [6568, 6487, 16]-code), using
(81−15, 81, 479398)-Net in Base 3 — Upper bound on s
There is no (66, 81, 479399)-net in base 3, because
- 1 times m-reduction [i] would yield (66, 80, 479399)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 147 808877 304461 461310 341494 045229 484307 > 380 [i]