Best Known (49, 65, s)-Nets in Base 4
(49, 65, 1028)-Net over F4 — Constructive and digital
Digital (49, 65, 1028)-net over F4, using
- 41 times duplication [i] based on digital (48, 64, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 16, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 16, 257)-net over F256, using
(49, 65, 1058)-Net over F4 — Digital
Digital (49, 65, 1058)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(465, 1058, F4, 16) (dual of [1058, 993, 17]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(461, 1033, F4, 16) (dual of [1033, 972, 17]-code), using
- construction XX applied to C1 = C([1022,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([1022,14]) [i] based on
- linear OA(456, 1023, F4, 15) (dual of [1023, 967, 16]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(456, 1023, F4, 15) (dual of [1023, 967, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(461, 1023, F4, 16) (dual of [1023, 962, 17]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(451, 1023, F4, 14) (dual of [1023, 972, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([1022,14]) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(461, 1033, F4, 16) (dual of [1033, 972, 17]-code), using
(49, 65, 97786)-Net in Base 4 — Upper bound on s
There is no (49, 65, 97787)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1361 186345 006612 205100 881433 460445 638695 > 465 [i]