Best Known (233−33, 233, s)-Nets in Base 3
(233−33, 233, 3693)-Net over F3 — Constructive and digital
Digital (200, 233, 3693)-net over F3, using
- 34 times duplication [i] based on digital (196, 229, 3693)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 3693, F3, 33, 33) (dual of [(3693, 33), 121640, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3229, 59089, F3, 33) (dual of [59089, 58860, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 59090, F3, 33) (dual of [59090, 58861, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 59090, F3, 33) (dual of [59090, 58861, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3229, 59089, F3, 33) (dual of [59089, 58860, 34]-code), using
- net defined by OOA [i] based on linear OOA(3229, 3693, F3, 33, 33) (dual of [(3693, 33), 121640, 34]-NRT-code), using
(233−33, 233, 28389)-Net over F3 — Digital
Digital (200, 233, 28389)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 28389, F3, 2, 33) (dual of [(28389, 2), 56545, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3233, 29551, F3, 2, 33) (dual of [(29551, 2), 58869, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3233, 59102, F3, 33) (dual of [59102, 58869, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(3221, 59050, F3, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3181, 59050, F3, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(312, 52, F3, 5) (dual of [52, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- OOA 2-folding [i] based on linear OA(3233, 59102, F3, 33) (dual of [59102, 58869, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3233, 29551, F3, 2, 33) (dual of [(29551, 2), 58869, 34]-NRT-code), using
(233−33, 233, large)-Net in Base 3 — Upper bound on s
There is no (200, 233, large)-net in base 3, because
- 31 times m-reduction [i] would yield (200, 202, large)-net in base 3, but