Best Known (80−19, 80, s)-Nets in Base 4
(80−19, 80, 1032)-Net over F4 — Constructive and digital
Digital (61, 80, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 20, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
(80−19, 80, 1219)-Net over F4 — Digital
Digital (61, 80, 1219)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(480, 1219, F4, 19) (dual of [1219, 1139, 20]-code), using
- 177 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 16 times 0, 1, 29 times 0, 1, 46 times 0, 1, 67 times 0) [i] based on linear OA(471, 1033, F4, 19) (dual of [1033, 962, 20]-code), using
- construction XX applied to C1 = C([1022,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1022,17]) [i] based on
- linear OA(466, 1023, F4, 18) (dual of [1023, 957, 19]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [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(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(461, 1023, F4, 17) (dual of [1023, 962, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([1022,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([1022,17]) [i] based on
- 177 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 16 times 0, 1, 29 times 0, 1, 46 times 0, 1, 67 times 0) [i] based on linear OA(471, 1033, F4, 19) (dual of [1033, 962, 20]-code), using
(80−19, 80, 266297)-Net in Base 4 — Upper bound on s
There is no (61, 80, 266298)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 79, 266298)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 365377 789391 961190 933311 175236 958162 264212 245101 > 479 [i]