Best Known (202−31, 202, s)-Nets in Base 3
(202−31, 202, 3937)-Net over F3 — Constructive and digital
Digital (171, 202, 3937)-net over F3, using
- net defined by OOA [i] based on linear OOA(3202, 3937, F3, 31, 31) (dual of [(3937, 31), 121845, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3202, 59056, F3, 31) (dual of [59056, 58854, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3202, 59056, F3, 31) (dual of [59056, 58854, 32]-code), using
(202−31, 202, 17919)-Net over F3 — Digital
Digital (171, 202, 17919)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 17919, F3, 3, 31) (dual of [(17919, 3), 53555, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3202, 19686, F3, 3, 31) (dual of [(19686, 3), 58856, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3202, 59058, F3, 31) (dual of [59058, 58856, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 59060, F3, 31) (dual of [59060, 58858, 32]-code), using
- OOA 3-folding [i] based on linear OA(3202, 59058, F3, 31) (dual of [59058, 58856, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3202, 19686, F3, 3, 31) (dual of [(19686, 3), 58856, 32]-NRT-code), using
(202−31, 202, 7946221)-Net in Base 3 — Upper bound on s
There is no (171, 202, 7946222)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 201, 7946222)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 796842 745141 224460 230477 511435 915709 214969 959287 266504 076482 860763 016817 704082 329047 045950 800041 > 3201 [i]