Best Known (138−15, 138, s)-Nets in Base 3
(138−15, 138, 227767)-Net over F3 — Constructive and digital
Digital (123, 138, 227767)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (115, 130, 227760)-net over F3, using
- net defined by OOA [i] based on linear OOA(3130, 227760, F3, 15, 15) (dual of [(227760, 15), 3416270, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3130, 1594321, F3, 15) (dual of [1594321, 1594191, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3130, 1594322, F3, 15) (dual of [1594322, 1594192, 16]-code), using
- 1 times truncation [i] based on linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3130, 1594322, F3, 15) (dual of [1594322, 1594192, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3130, 1594321, F3, 15) (dual of [1594321, 1594191, 16]-code), using
- net defined by OOA [i] based on linear OOA(3130, 227760, F3, 15, 15) (dual of [(227760, 15), 3416270, 16]-NRT-code), using
- digital (1, 8, 7)-net over F3, using
(138−15, 138, 675614)-Net over F3 — Digital
Digital (123, 138, 675614)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 675614, F3, 2, 15) (dual of [(675614, 2), 1351090, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 797184, F3, 2, 15) (dual of [(797184, 2), 1594230, 16]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3137, 797184, F3, 2, 15) (dual of [(797184, 2), 1594231, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- 31 times duplication [i] based on linear OOA(3137, 797184, F3, 2, 15) (dual of [(797184, 2), 1594231, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 797184, F3, 2, 15) (dual of [(797184, 2), 1594230, 16]-NRT-code), using
(138−15, 138, large)-Net in Base 3 — Upper bound on s
There is no (123, 138, large)-net in base 3, because
- 13 times m-reduction [i] would yield (123, 125, large)-net in base 3, but