Best Known (193, 193+26, s)-Nets in Base 3
(193, 193+26, 40888)-Net over F3 — Constructive and digital
Digital (193, 219, 40888)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (179, 205, 40881)-net over F3, using
- net defined by OOA [i] based on linear OOA(3205, 40881, F3, 26, 26) (dual of [(40881, 26), 1062701, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- OA 13-folding and stacking [i] based on linear OA(3205, 531453, F3, 26) (dual of [531453, 531248, 27]-code), using
- net defined by OOA [i] based on linear OOA(3205, 40881, F3, 26, 26) (dual of [(40881, 26), 1062701, 27]-NRT-code), using
- digital (1, 14, 7)-net over F3, using
(193, 193+26, 177167)-Net over F3 — Digital
Digital (193, 219, 177167)-net over F3, using
- 32 times duplication [i] based on digital (191, 217, 177167)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 177167, F3, 3, 26) (dual of [(177167, 3), 531284, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3217, 531501, F3, 26) (dual of [531501, 531284, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3216, 531500, F3, 26) (dual of [531500, 531284, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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 Ce(25) ⊂ Ce(19) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3216, 531500, F3, 26) (dual of [531500, 531284, 27]-code), using
- OOA 3-folding [i] based on linear OA(3217, 531501, F3, 26) (dual of [531501, 531284, 27]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 177167, F3, 3, 26) (dual of [(177167, 3), 531284, 27]-NRT-code), using
(193, 193+26, large)-Net in Base 3 — Upper bound on s
There is no (193, 219, large)-net in base 3, because
- 24 times m-reduction [i] would yield (193, 195, large)-net in base 3, but