Best Known (222−33, 222, s)-Nets in Base 3
(222−33, 222, 3691)-Net over F3 — Constructive and digital
Digital (189, 222, 3691)-net over F3, using
- 31 times duplication [i] based on digital (188, 221, 3691)-net over F3, using
- net defined by OOA [i] based on linear OOA(3221, 3691, F3, 33, 33) (dual of [(3691, 33), 121582, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3221, 59057, F3, 33) (dual of [59057, 58836, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(33) ⊂ Ce(31) [i] based on
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3221, 59057, F3, 33) (dual of [59057, 58836, 34]-code), using
- net defined by OOA [i] based on linear OOA(3221, 3691, F3, 33, 33) (dual of [(3691, 33), 121582, 34]-NRT-code), using
(222−33, 222, 19690)-Net over F3 — Digital
Digital (189, 222, 19690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3222, 19690, F3, 3, 33) (dual of [(19690, 3), 58848, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3222, 59070, F3, 33) (dual of [59070, 58848, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 59071, F3, 33) (dual of [59071, 58849, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3221, 59050, F3, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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 C([0,16]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 59071, F3, 33) (dual of [59071, 58849, 34]-code), using
- OOA 3-folding [i] based on linear OA(3222, 59070, F3, 33) (dual of [59070, 58848, 34]-code), using
(222−33, 222, large)-Net in Base 3 — Upper bound on s
There is no (189, 222, large)-net in base 3, because
- 31 times m-reduction [i] would yield (189, 191, large)-net in base 3, but