Best Known (30−7, 30, s)-Nets in Base 3
(30−7, 30, 731)-Net over F3 — Constructive and digital
Digital (23, 30, 731)-net over F3, using
- net defined by OOA [i] based on linear OOA(330, 731, F3, 7, 7) (dual of [(731, 7), 5087, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(330, 2194, F3, 7) (dual of [2194, 2164, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(330, 2195, F3, 7) (dual of [2195, 2165, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(330, 2195, F3, 7) (dual of [2195, 2165, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(330, 2194, F3, 7) (dual of [2194, 2164, 8]-code), using
(30−7, 30, 1098)-Net over F3 — Digital
Digital (23, 30, 1098)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(330, 1098, F3, 2, 7) (dual of [(1098, 2), 2166, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(330, 2196, F3, 7) (dual of [2196, 2166, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(38, 9, F3, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
- dual of repetition code with length 9 [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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(330, 2196, F3, 7) (dual of [2196, 2166, 8]-code), using
(30−7, 30, 37196)-Net in Base 3 — Upper bound on s
There is no (23, 30, 37197)-net in base 3, because
- 1 times m-reduction [i] would yield (23, 29, 37197)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 68 635695 972811 > 329 [i]