Best Known (84−20, 84, s)-Nets in Base 4
(84−20, 84, 1032)-Net over F4 — Constructive and digital
Digital (64, 84, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
(84−20, 84, 1233)-Net over F4 — Digital
Digital (64, 84, 1233)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(484, 1233, F4, 20) (dual of [1233, 1149, 21]-code), using
- 192 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 33 times 0, 1, 51 times 0, 1, 68 times 0) [i] based on linear OA(476, 1033, F4, 20) (dual of [1033, 957, 21]-code), using
- construction XX applied to C1 = C([1022,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([1022,18]) [i] based on
- linear OA(471, 1023, F4, 19) (dual of [1023, 952, 20]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(471, 1023, F4, 19) (dual of [1023, 952, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(476, 1023, F4, 20) (dual of [1023, 947, 21]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(466, 1023, F4, 18) (dual of [1023, 957, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [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([1022,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([1022,18]) [i] based on
- 192 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 33 times 0, 1, 51 times 0, 1, 68 times 0) [i] based on linear OA(476, 1033, F4, 20) (dual of [1033, 957, 21]-code), using
(84−20, 84, 172242)-Net in Base 4 — Upper bound on s
There is no (64, 84, 172243)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 374 164713 053285 795715 908278 783041 889769 170437 349030 > 484 [i]