Best Known (63−20, 63, s)-Nets in Base 4
(63−20, 63, 240)-Net over F4 — Constructive and digital
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
(63−20, 63, 282)-Net over F4 — Digital
Digital (43, 63, 282)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(463, 282, F4, 20) (dual of [282, 219, 21]-code), using
- 15 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 7 times 0) [i] based on linear OA(459, 263, F4, 20) (dual of [263, 204, 21]-code), using
- construction XX applied to C1 = C([254,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([254,18]) [i] based on
- 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(455, 255, F4, 19) (dual of [255, 200, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(459, 255, F4, 20) (dual of [255, 196, 21]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [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(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, 4, F4, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([254,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([254,18]) [i] based on
- 15 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 7 times 0) [i] based on linear OA(459, 263, F4, 20) (dual of [263, 204, 21]-code), using
(63−20, 63, 9364)-Net in Base 4 — Upper bound on s
There is no (43, 63, 9365)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 85 144630 934099 447758 931092 979771 052628 > 463 [i]