Best Known (223−30, 223, s)-Nets in Base 3
(223−30, 223, 11811)-Net over F3 — Constructive and digital
Digital (193, 223, 11811)-net over F3, using
- 31 times duplication [i] based on digital (192, 222, 11811)-net over F3, using
- net defined by OOA [i] based on linear OOA(3222, 11811, F3, 30, 30) (dual of [(11811, 30), 354108, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3222, 177165, F3, 30) (dual of [177165, 176943, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 177170, F3, 30) (dual of [177170, 176948, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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 Ce(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 177170, F3, 30) (dual of [177170, 176948, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(3222, 177165, F3, 30) (dual of [177165, 176943, 31]-code), using
- net defined by OOA [i] based on linear OOA(3222, 11811, F3, 30, 30) (dual of [(11811, 30), 354108, 31]-NRT-code), using
(223−30, 223, 57445)-Net over F3 — Digital
Digital (193, 223, 57445)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3223, 57445, F3, 3, 30) (dual of [(57445, 3), 172112, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 59057, F3, 3, 30) (dual of [(59057, 3), 176948, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3223, 177171, F3, 30) (dual of [177171, 176948, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3222, 177170, F3, 30) (dual of [177170, 176948, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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 Ce(30) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3222, 177170, F3, 30) (dual of [177170, 176948, 31]-code), using
- OOA 3-folding [i] based on linear OA(3223, 177171, F3, 30) (dual of [177171, 176948, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(3223, 59057, F3, 3, 30) (dual of [(59057, 3), 176948, 31]-NRT-code), using
(223−30, 223, large)-Net in Base 3 — Upper bound on s
There is no (193, 223, large)-net in base 3, because
- 28 times m-reduction [i] would yield (193, 195, large)-net in base 3, but