Best Known (175, 175+27, s)-Nets in Base 3
(175, 175+27, 13628)-Net over F3 — Constructive and digital
Digital (175, 202, 13628)-net over F3, using
- 32 times duplication [i] based on digital (173, 200, 13628)-net over F3, using
- net defined by OOA [i] based on linear OOA(3200, 13628, F3, 27, 27) (dual of [(13628, 27), 367756, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3200, 177165, F3, 27) (dual of [177165, 176965, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 177171, F3, 27) (dual of [177171, 176971, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3199, 177148, F3, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3200, 177171, F3, 27) (dual of [177171, 176971, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3200, 177165, F3, 27) (dual of [177165, 176965, 28]-code), using
- net defined by OOA [i] based on linear OOA(3200, 13628, F3, 27, 27) (dual of [(13628, 27), 367756, 28]-NRT-code), using
(175, 175+27, 59057)-Net over F3 — Digital
Digital (175, 202, 59057)-net over F3, using
- 31 times duplication [i] based on digital (174, 201, 59057)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3201, 59057, F3, 3, 27) (dual of [(59057, 3), 176970, 28]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3200, 59057, F3, 3, 27) (dual of [(59057, 3), 176971, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3200, 177171, F3, 27) (dual of [177171, 176971, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3199, 177148, F3, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,13]) ⊂ C([0,12]) [i] based on
- OOA 3-folding [i] based on linear OA(3200, 177171, F3, 27) (dual of [177171, 176971, 28]-code), using
- 31 times duplication [i] based on linear OOA(3200, 59057, F3, 3, 27) (dual of [(59057, 3), 176971, 28]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3201, 59057, F3, 3, 27) (dual of [(59057, 3), 176970, 28]-NRT-code), using
(175, 175+27, large)-Net in Base 3 — Upper bound on s
There is no (175, 202, large)-net in base 3, because
- 25 times m-reduction [i] would yield (175, 177, large)-net in base 3, but