Best Known (203−24, 203, s)-Nets in Base 3
(203−24, 203, 44291)-Net over F3 — Constructive and digital
Digital (179, 203, 44291)-net over F3, using
- 1 times m-reduction [i] based on digital (179, 204, 44291)-net over F3, using
- net defined by OOA [i] based on linear OOA(3204, 44291, F3, 25, 25) (dual of [(44291, 25), 1107071, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3204, 531493, F3, 25) (dual of [531493, 531289, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 531501, F3, 25) (dual of [531501, 531297, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3193, 531442, F3, 25) (dual of [531442, 531249, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(311, 59, F3, 5) (dual of [59, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3204, 531501, F3, 25) (dual of [531501, 531297, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3204, 531493, F3, 25) (dual of [531493, 531289, 26]-code), using
- net defined by OOA [i] based on linear OOA(3204, 44291, F3, 25, 25) (dual of [(44291, 25), 1107071, 26]-NRT-code), using
(203−24, 203, 177166)-Net over F3 — Digital
Digital (179, 203, 177166)-net over F3, using
- 31 times duplication [i] based on digital (178, 202, 177166)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 177166, F3, 3, 24) (dual of [(177166, 3), 531296, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3202, 531498, F3, 24) (dual of [531498, 531296, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 531499, F3, 24) (dual of [531499, 531297, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3193, 531442, F3, 25) (dual of [531442, 531249, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 531499, F3, 24) (dual of [531499, 531297, 25]-code), using
- OOA 3-folding [i] based on linear OA(3202, 531498, F3, 24) (dual of [531498, 531296, 25]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 177166, F3, 3, 24) (dual of [(177166, 3), 531296, 25]-NRT-code), using
(203−24, 203, large)-Net in Base 3 — Upper bound on s
There is no (179, 203, large)-net in base 3, because
- 22 times m-reduction [i] would yield (179, 181, large)-net in base 3, but