Best Known (62−11, 62, s)-Nets in Base 3
(62−11, 62, 1317)-Net over F3 — Constructive and digital
Digital (51, 62, 1317)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (46, 57, 1313)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 1313, F3, 11, 11) (dual of [(1313, 11), 14386, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(357, 6566, F3, 11) (dual of [6566, 6509, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(357, 6566, F3, 11) (dual of [6566, 6509, 12]-code), using
- net defined by OOA [i] based on linear OOA(357, 1313, F3, 11, 11) (dual of [(1313, 11), 14386, 12]-NRT-code), using
- digital (0, 5, 4)-net over F3, using
(62−11, 62, 3545)-Net over F3 — Digital
Digital (51, 62, 3545)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(362, 3545, F3, 11) (dual of [3545, 3483, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(362, 6567, F3, 11) (dual of [6567, 6505, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- dual of repetition code with length 6 [i]
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(362, 6567, F3, 11) (dual of [6567, 6505, 12]-code), using
(62−11, 62, 862349)-Net in Base 3 — Upper bound on s
There is no (51, 62, 862350)-net in base 3, because
- 1 times m-reduction [i] would yield (51, 61, 862350)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 127173 874307 368299 530442 607541 > 361 [i]