Best Known (61, 61+16, s)-Nets in Base 3
(61, 61+16, 464)-Net over F3 — Constructive and digital
Digital (61, 77, 464)-net over F3, using
- 31 times duplication [i] based on digital (60, 76, 464)-net over F3, using
- t-expansion [i] based on digital (59, 76, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 19, 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, 19, 116)-net over F81, using
- t-expansion [i] based on digital (59, 76, 464)-net over F3, using
(61, 61+16, 1164)-Net over F3 — Digital
Digital (61, 77, 1164)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(377, 1164, F3, 16) (dual of [1164, 1087, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(377, 2208, F3, 16) (dual of [2208, 2131, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- 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(357, 2187, F3, 13) (dual of [2187, 2130, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(34, 19, F3, 2) (dual of [19, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(377, 2208, F3, 16) (dual of [2208, 2131, 17]-code), using
(61, 61+16, 73605)-Net in Base 3 — Upper bound on s
There is no (61, 77, 73606)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 474937 464082 505854 293905 626027 825105 > 377 [i]