Best Known (21−6, 21, s)-Nets in Base 3
(21−6, 21, 114)-Net over F3 — Constructive and digital
Digital (15, 21, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 7, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
(21−6, 21, 252)-Net over F3 — Digital
Digital (15, 21, 252)-net over F3, using
- net defined by OOA [i] based on linear OOA(321, 252, F3, 6, 6) (dual of [(252, 6), 1491, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(321, 252, F3, 5, 6) (dual of [(252, 5), 1239, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(321, 252, F3, 6) (dual of [252, 231, 7]-code), using
- construction XX applied to C1 = C([241,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([241,4]) [i] based on
- linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(321, 242, F3, 6) (dual of [242, 221, 7]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(311, 242, F3, 4) (dual of [242, 231, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([241,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([241,4]) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(321, 252, F3, 6) (dual of [252, 231, 7]-code), using
- appending kth column [i] based on linear OOA(321, 252, F3, 5, 6) (dual of [(252, 5), 1239, 7]-NRT-code), using
(21−6, 21, 1984)-Net in Base 3 — Upper bound on s
There is no (15, 21, 1985)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10467 893531 > 321 [i]