Best Known (51, 51+15, s)-Nets in Base 4
(51, 51+15, 1032)-Net over F4 — Constructive and digital
Digital (51, 66, 1032)-net over F4, using
- 42 times duplication [i] based on digital (49, 64, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 16, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 16, 258)-net over F256, using
(51, 51+15, 1404)-Net over F4 — Digital
Digital (51, 66, 1404)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(466, 1404, F4, 15) (dual of [1404, 1338, 16]-code), using
- 361 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0, 1, 40 times 0, 1, 64 times 0, 1, 92 times 0, 1, 116 times 0) [i] based on linear OA(456, 1033, F4, 15) (dual of [1033, 977, 16]-code), using
- construction XX applied to C1 = C([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- linear OA(451, 1023, F4, 14) (dual of [1023, 972, 15]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(451, 1023, F4, 14) (dual of [1023, 972, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(456, 1023, F4, 15) (dual of [1023, 967, 16]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(446, 1023, F4, 13) (dual of [1023, 977, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [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([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- 361 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0, 1, 40 times 0, 1, 64 times 0, 1, 92 times 0, 1, 116 times 0) [i] based on linear OA(456, 1033, F4, 15) (dual of [1033, 977, 16]-code), using
(51, 51+15, 438883)-Net in Base 4 — Upper bound on s
There is no (51, 66, 438884)-net in base 4, because
- 1 times m-reduction [i] would yield (51, 65, 438884)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1361 141978 968661 108404 188426 764271 057160 > 465 [i]