Best Known (42−6, 42, s)-Nets in Base 3
(42−6, 42, 19690)-Net over F3 — Constructive and digital
Digital (36, 42, 19690)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 19690, F3, 6, 6) (dual of [(19690, 6), 118098, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(342, 19690, F3, 5, 6) (dual of [(19690, 5), 98408, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(342, 59070, F3, 6) (dual of [59070, 59028, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(321, 59049, F3, 4) (dual of [59049, 59028, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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 Ce(6) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(342, 59070, F3, 6) (dual of [59070, 59028, 7]-code), using
- appending kth column [i] based on linear OOA(342, 19690, F3, 5, 6) (dual of [(19690, 5), 98408, 7]-NRT-code), using
(42−6, 42, 59071)-Net over F3 — Digital
Digital (36, 42, 59071)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 59071, F3, 6, 6) (dual of [(59071, 6), 354384, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(342, 59071, F3, 5, 6) (dual of [(59071, 5), 295313, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(342, 59071, F3, 6) (dual of [59071, 59029, 7]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(321, 59049, F3, 4) (dual of [59049, 59028, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(342, 59071, F3, 6) (dual of [59071, 59029, 7]-code), using
- appending kth column [i] based on linear OOA(342, 59071, F3, 5, 6) (dual of [(59071, 5), 295313, 7]-NRT-code), using
(42−6, 42, 4345613)-Net in Base 3 — Upper bound on s
There is no (36, 42, 4345614)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 109 419047 114906 553025 > 342 [i]