Best Known (207−25, 207, s)-Nets in Base 3
(207−25, 207, 44294)-Net over F3 — Constructive and digital
Digital (182, 207, 44294)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (168, 193, 44286)-net over F3, using
- net defined by OOA [i] based on linear OOA(3193, 44286, F3, 25, 25) (dual of [(44286, 25), 1106957, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3193, 531433, F3, 25) (dual of [531433, 531240, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3193, 531433, F3, 25) (dual of [531433, 531240, 26]-code), using
- net defined by OOA [i] based on linear OOA(3193, 44286, F3, 25, 25) (dual of [(44286, 25), 1106957, 26]-NRT-code), using
- digital (2, 14, 8)-net over F3, using
(207−25, 207, 177168)-Net over F3 — Digital
Digital (182, 207, 177168)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3207, 177168, F3, 3, 25) (dual of [(177168, 3), 531297, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3204, 177167, F3, 3, 25) (dual of [(177167, 3), 531297, 26]-NRT-code), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on linear OA(3204, 531501, F3, 25) (dual of [531501, 531297, 26]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3204, 177167, F3, 3, 25) (dual of [(177167, 3), 531297, 26]-NRT-code), using
(207−25, 207, large)-Net in Base 3 — Upper bound on s
There is no (182, 207, large)-net in base 3, because
- 23 times m-reduction [i] would yield (182, 184, large)-net in base 3, but