Best Known (164−20, 164, s)-Nets in Base 3
(164−20, 164, 53148)-Net over F3 — Constructive and digital
Digital (144, 164, 53148)-net over F3, using
- 31 times duplication [i] based on digital (143, 163, 53148)-net over F3, using
- net defined by OOA [i] based on linear OOA(3163, 53148, F3, 20, 20) (dual of [(53148, 20), 1062797, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3163, 531480, F3, 20) (dual of [531480, 531317, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3163, 531483, F3, 20) (dual of [531483, 531320, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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 Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3163, 531483, F3, 20) (dual of [531483, 531320, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3163, 531480, F3, 20) (dual of [531480, 531317, 21]-code), using
- net defined by OOA [i] based on linear OOA(3163, 53148, F3, 20, 20) (dual of [(53148, 20), 1062797, 21]-NRT-code), using
(164−20, 164, 177161)-Net over F3 — Digital
Digital (144, 164, 177161)-net over F3, using
- 31 times duplication [i] based on digital (143, 163, 177161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 177161, F3, 3, 20) (dual of [(177161, 3), 531320, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3163, 531483, F3, 20) (dual of [531483, 531320, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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 Ce(19) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3163, 531483, F3, 20) (dual of [531483, 531320, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 177161, F3, 3, 20) (dual of [(177161, 3), 531320, 21]-NRT-code), using
(164−20, 164, large)-Net in Base 3 — Upper bound on s
There is no (144, 164, large)-net in base 3, because
- 18 times m-reduction [i] would yield (144, 146, large)-net in base 3, but