Best Known (74, 98, s)-Nets in Base 4
(74, 98, 1028)-Net over F4 — Constructive and digital
Digital (74, 98, 1028)-net over F4, using
- 42 times duplication [i] based on digital (72, 96, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
(74, 98, 1177)-Net over F4 — Digital
Digital (74, 98, 1177)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(498, 1177, F4, 24) (dual of [1177, 1079, 25]-code), using
- 137 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 37 times 0, 1, 52 times 0) [i] based on linear OA(491, 1033, F4, 24) (dual of [1033, 942, 25]-code), using
- construction XX applied to C1 = C([1022,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([1022,22]) [i] based on
- linear OA(486, 1023, F4, 23) (dual of [1023, 937, 24]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(486, 1023, F4, 23) (dual of [1023, 937, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(491, 1023, F4, 24) (dual of [1023, 932, 25]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [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([1022,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([1022,22]) [i] based on
- 137 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 37 times 0, 1, 52 times 0) [i] based on linear OA(491, 1033, F4, 24) (dual of [1033, 942, 25]-code), using
(74, 98, 145557)-Net in Base 4 — Upper bound on s
There is no (74, 98, 145558)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 100436 406314 667271 815341 897190 786157 197031 300624 723564 137370 > 498 [i]