Best Known (213−46, 213, s)-Nets in Base 3
(213−46, 213, 688)-Net over F3 — Constructive and digital
Digital (167, 213, 688)-net over F3, using
- 31 times duplication [i] based on digital (166, 212, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 53, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 53, 172)-net over F81, using
(213−46, 213, 1677)-Net over F3 — Digital
Digital (167, 213, 1677)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3213, 1677, F3, 46) (dual of [1677, 1464, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 2197, F3, 46) (dual of [2197, 1984, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(45) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 2197, F3, 46) (dual of [2197, 1984, 47]-code), using
(213−46, 213, 123567)-Net in Base 3 — Upper bound on s
There is no (167, 213, 123568)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 423498 996575 636516 504046 958042 122674 740468 732913 717527 907133 989010 360865 223511 552691 797106 443531 716673 > 3213 [i]