Best Known (68, 87, s)-Nets in Base 3
(68, 87, 464)-Net over F3 — Constructive and digital
Digital (68, 87, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (68, 88, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 22, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 22, 116)-net over F81, using
(68, 87, 1098)-Net over F3 — Digital
Digital (68, 87, 1098)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(387, 1098, F3, 2, 19) (dual of [(1098, 2), 2109, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(387, 2196, F3, 19) (dual of [2196, 2109, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(387, 2197, F3, 19) (dual of [2197, 2110, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(385, 2187, F3, 19) (dual of [2187, 2102, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(371, 2187, F3, 16) (dual of [2187, 2116, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(387, 2197, F3, 19) (dual of [2197, 2110, 20]-code), using
- OOA 2-folding [i] based on linear OA(387, 2196, F3, 19) (dual of [2196, 2109, 20]-code), using
(68, 87, 75133)-Net in Base 3 — Upper bound on s
There is no (68, 87, 75134)-net in base 3, because
- 1 times m-reduction [i] would yield (68, 86, 75134)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 107764 983579 360063 624346 904120 830868 319085 > 386 [i]