Best Known (18, 23, s)-Nets in Base 3
(18, 23, 1097)-Net over F3 — Constructive and digital
Digital (18, 23, 1097)-net over F3, using
- net defined by OOA [i] based on linear OOA(323, 1097, F3, 5, 5) (dual of [(1097, 5), 5462, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(323, 2195, F3, 5) (dual of [2195, 2172, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 2196, F3, 5) (dual of [2196, 2173, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(323, 2196, F3, 5) (dual of [2196, 2173, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(323, 2195, F3, 5) (dual of [2195, 2172, 6]-code), using
(18, 23, 2197)-Net over F3 — Digital
Digital (18, 23, 2197)-net over F3, using
- net defined by OOA [i] based on linear OOA(323, 2197, F3, 5, 5) (dual of [(2197, 5), 10962, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(323, 2197, F3, 4, 5) (dual of [(2197, 4), 8765, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(323, 2197, F3, 5) (dual of [2197, 2174, 6]-code), using
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 2 step Varšamov–Edel lengthening with (ri) = (1, 0) [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(323, 2197, F3, 5) (dual of [2197, 2174, 6]-code), using
- appending kth column [i] based on linear OOA(323, 2197, F3, 4, 5) (dual of [(2197, 4), 8765, 6]-NRT-code), using
(18, 23, 125260)-Net in Base 3 — Upper bound on s
There is no (18, 23, 125261)-net in base 3, because
- 1 times m-reduction [i] would yield (18, 22, 125261)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 31381 387809 > 322 [i]