Best Known (48−23, 48, s)-Nets in Base 64
(48−23, 48, 373)-Net over F64 — Constructive and digital
Digital (25, 48, 373)-net over F64, using
- 641 times duplication [i] based on digital (24, 47, 373)-net over F64, using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
(48−23, 48, 516)-Net in Base 64 — Constructive
(25, 48, 516)-net in base 64, using
- base change [i] based on digital (13, 36, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 24, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 12, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(48−23, 48, 1871)-Net over F64 — Digital
Digital (25, 48, 1871)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6448, 1871, F64, 2, 23) (dual of [(1871, 2), 3694, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6448, 2054, F64, 2, 23) (dual of [(2054, 2), 4060, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6448, 4108, F64, 23) (dual of [4108, 4060, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(6448, 4108, F64, 23) (dual of [4108, 4060, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(6448, 2054, F64, 2, 23) (dual of [(2054, 2), 4060, 24]-NRT-code), using
(48−23, 48, 4064316)-Net in Base 64 — Upper bound on s
There is no (25, 48, 4064317)-net in base 64, because
- 1 times m-reduction [i] would yield (25, 47, 4064317)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 7 770692 433785 365238 503305 493012 640661 585907 137766 388582 877393 495148 850186 377589 617420 > 6447 [i]