Best Known (237−13, 237, s)-Nets in Base 3
(237−13, 237, 5612086)-Net over F3 — Constructive and digital
Digital (224, 237, 5612086)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (35, 41, 19686)-net over F3, using
- net defined by OOA [i] based on linear OOA(341, 19686, F3, 6, 6) (dual of [(19686, 6), 118075, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(341, 19686, F3, 5, 6) (dual of [(19686, 5), 98389, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(341, 59058, F3, 6) (dual of [59058, 59017, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [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(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(341, 59058, F3, 6) (dual of [59058, 59017, 7]-code), using
- appending kth column [i] based on linear OOA(341, 19686, F3, 5, 6) (dual of [(19686, 5), 98389, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(341, 19686, F3, 6, 6) (dual of [(19686, 6), 118075, 7]-NRT-code), using
- digital (183, 196, 5592400)-net over F3, using
- trace code for nets [i] based on digital (85, 98, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- trace code for nets [i] based on digital (85, 98, 2796200)-net over F9, using
- digital (35, 41, 19686)-net over F3, using
(237−13, 237, large)-Net over F3 — Digital
Digital (224, 237, large)-net over F3, using
- t-expansion [i] based on digital (221, 237, large)-net over F3, using
- 4 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
- 4 times m-reduction [i] based on digital (221, 241, large)-net over F3, using
(237−13, 237, large)-Net in Base 3 — Upper bound on s
There is no (224, 237, large)-net in base 3, because
- 11 times m-reduction [i] would yield (224, 226, large)-net in base 3, but