Best Known (53, 53+16, s)-Nets in Base 4
(53, 53+16, 1032)-Net over F4 — Constructive and digital
Digital (53, 69, 1032)-net over F4, using
- 41 times duplication [i] based on digital (52, 68, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 17, 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, 17, 258)-net over F256, using
(53, 53+16, 1283)-Net over F4 — Digital
Digital (53, 69, 1283)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(469, 1283, F4, 16) (dual of [1283, 1214, 17]-code), using
- 242 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 41 times 0, 1, 64 times 0, 1, 89 times 0) [i] based on linear OA(461, 1033, F4, 16) (dual of [1033, 972, 17]-code), using
- construction XX applied to C1 = C([1022,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([1022,14]) [i] based on
- 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(456, 1023, F4, 15) (dual of [1023, 967, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(461, 1023, F4, 16) (dual of [1023, 962, 17]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [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(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([1022,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([1022,14]) [i] based on
- 242 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 41 times 0, 1, 64 times 0, 1, 89 times 0) [i] based on linear OA(461, 1033, F4, 16) (dual of [1033, 972, 17]-code), using
(53, 53+16, 195579)-Net in Base 4 — Upper bound on s
There is no (53, 69, 195580)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 348461 332108 122018 141044 584332 964896 997519 > 469 [i]