Best Known (43, 43+19, s)-Nets in Base 4
(43, 43+19, 240)-Net over F4 — Constructive and digital
Digital (43, 62, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (43, 63, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 21, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 21, 80)-net over F64, using
(43, 43+19, 311)-Net over F4 — Digital
Digital (43, 62, 311)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(462, 311, F4, 19) (dual of [311, 249, 20]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 11 times 0, 1, 16 times 0) [i] based on linear OA(455, 261, F4, 19) (dual of [261, 206, 20]-code), using
- construction XX applied to C1 = C([254,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([254,17]) [i] based on
- linear OA(453, 255, F4, 18) (dual of [255, 202, 19]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(451, 255, F4, 18) (dual of [255, 204, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(455, 255, F4, 19) (dual of [255, 200, 20]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(449, 255, F4, 17) (dual of [255, 206, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 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, 2, F4, 0) (dual of [2, 2, 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 (see above)
- construction XX applied to C1 = C([254,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([254,17]) [i] based on
- 43 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 11 times 0, 1, 16 times 0) [i] based on linear OA(455, 261, F4, 19) (dual of [261, 206, 20]-code), using
(43, 43+19, 16637)-Net in Base 4 — Upper bound on s
There is no (43, 62, 16638)-net in base 4, because
- 1 times m-reduction [i] would yield (43, 61, 16638)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5 319639 257519 530883 781783 021890 051695 > 461 [i]