Best Known (41, 61, s)-Nets in Base 64
(41, 61, 26215)-Net over F64 — Constructive and digital
Digital (41, 61, 26215)-net over F64, using
- 1 times m-reduction [i] based on digital (41, 62, 26215)-net over F64, using
- net defined by OOA [i] based on linear OOA(6462, 26215, F64, 21, 21) (dual of [(26215, 21), 550453, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6462, 262151, F64, 21) (dual of [262151, 262089, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 262152, F64, 21) (dual of [262152, 262090, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 262152, F64, 21) (dual of [262152, 262090, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6462, 262151, F64, 21) (dual of [262151, 262089, 22]-code), using
- net defined by OOA [i] based on linear OOA(6462, 26215, F64, 21, 21) (dual of [(26215, 21), 550453, 22]-NRT-code), using
(41, 61, 131079)-Net over F64 — Digital
Digital (41, 61, 131079)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6461, 131079, F64, 2, 20) (dual of [(131079, 2), 262097, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6461, 262158, F64, 20) (dual of [262158, 262097, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, 262159, F64, 20) (dual of [262159, 262098, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-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 Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6461, 262159, F64, 20) (dual of [262159, 262098, 21]-code), using
- OOA 2-folding [i] based on linear OA(6461, 262158, F64, 20) (dual of [262158, 262097, 21]-code), using
(41, 61, large)-Net in Base 64 — Upper bound on s
There is no (41, 61, large)-net in base 64, because
- 18 times m-reduction [i] would yield (41, 43, large)-net in base 64, but