Best Known (62−27, 62, s)-Nets in Base 64
(62−27, 62, 513)-Net over F64 — Constructive and digital
Digital (35, 62, 513)-net over F64, using
- t-expansion [i] based on digital (28, 62, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(62−27, 62, 1260)-Net in Base 64 — Constructive
(35, 62, 1260)-net in base 64, using
- net defined by OOA [i] based on OOA(6462, 1260, S64, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6462, 16381, S64, 27), using
- discarding factors based on OA(6462, 16386, S64, 27), using
- discarding parts of the base [i] based on linear OA(12853, 16386, F128, 27) (dual of [16386, 16333, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding parts of the base [i] based on linear OA(12853, 16386, F128, 27) (dual of [16386, 16333, 28]-code), using
- discarding factors based on OA(6462, 16386, S64, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6462, 16381, S64, 27), using
(62−27, 62, 4113)-Net over F64 — Digital
Digital (35, 62, 4113)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6462, 4113, F64, 27) (dual of [4113, 4051, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 4126, F64, 27) (dual of [4126, 4064, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,8]) [i] based on
- linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(649, 29, F64, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,64)), using
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- Reed–Solomon code RS(55,64) [i]
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- construction X applied to C([0,13]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 4126, F64, 27) (dual of [4126, 4064, 28]-code), using
(62−27, 62, large)-Net in Base 64 — Upper bound on s
There is no (35, 62, large)-net in base 64, because
- 25 times m-reduction [i] would yield (35, 37, large)-net in base 64, but