Best Known (93, 93+24, s)-Nets in Base 3
(93, 93+24, 640)-Net over F3 — Constructive and digital
Digital (93, 117, 640)-net over F3, using
- 31 times duplication [i] based on digital (92, 116, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 29, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 29, 160)-net over F81, using
(93, 93+24, 1464)-Net over F3 — Digital
Digital (93, 117, 1464)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3117, 1464, F3, 24) (dual of [1464, 1347, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3117, 2207, F3, 24) (dual of [2207, 2090, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3117, 2207, F3, 24) (dual of [2207, 2090, 25]-code), using
(93, 93+24, 118637)-Net in Base 3 — Upper bound on s
There is no (93, 117, 118638)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 66 560091 783328 830817 655914 733802 114854 155723 663584 126169 > 3117 [i]