Best Known (83−21, 83, s)-Nets in Base 4
(83−21, 83, 514)-Net over F4 — Constructive and digital
Digital (62, 83, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(41,16) in PG(82,4)) for nets [i] based on digital (21, 42, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
(83−21, 83, 1034)-Net over F4 — Digital
Digital (62, 83, 1034)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(483, 1034, F4, 21) (dual of [1034, 951, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(483, 1047, F4, 21) (dual of [1047, 964, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(461, 1024, F4, 17) (dual of [1024, 963, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(456, 1024, F4, 15) (dual of [1024, 968, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(45, 21, F4, 3) (dual of [21, 16, 4]-code or 21-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(483, 1047, F4, 21) (dual of [1047, 964, 22]-code), using
(83−21, 83, 130533)-Net in Base 4 — Upper bound on s
There is no (62, 83, 130534)-net in base 4, because
- 1 times m-reduction [i] would yield (62, 82, 130534)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 23 385495 947987 591858 594205 029149 463238 101456 698120 > 482 [i]