Best Known (254−136, 254, s)-Nets in Base 2
(254−136, 254, 57)-Net over F2 — Constructive and digital
Digital (118, 254, 57)-net over F2, using
- t-expansion [i] based on digital (110, 254, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
(254−136, 254, 73)-Net over F2 — Digital
Digital (118, 254, 73)-net over F2, using
- t-expansion [i] based on digital (114, 254, 73)-net over F2, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 114 and N(F) ≥ 73, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
(254−136, 254, 247)-Net over F2 — Upper bound on s (digital)
There is no digital (118, 254, 248)-net over F2, because
- 16 times m-reduction [i] would yield digital (118, 238, 248)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2238, 248, F2, 120) (dual of [248, 10, 121]-code), but
- residual code [i] would yield linear OA(2118, 127, F2, 60) (dual of [127, 9, 61]-code), but
- residual code [i] would yield linear OA(258, 66, F2, 30) (dual of [66, 8, 31]-code), but
- adding a parity check bit [i] would yield linear OA(259, 67, F2, 31) (dual of [67, 8, 32]-code), but
- “DHM†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(259, 67, F2, 31) (dual of [67, 8, 32]-code), but
- residual code [i] would yield linear OA(258, 66, F2, 30) (dual of [66, 8, 31]-code), but
- residual code [i] would yield linear OA(2118, 127, F2, 60) (dual of [127, 9, 61]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2238, 248, F2, 120) (dual of [248, 10, 121]-code), but
(254−136, 254, 248)-Net in Base 2 — Upper bound on s
There is no (118, 254, 249)-net in base 2, because
- 12 times m-reduction [i] would yield (118, 242, 249)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2242, 249, S2, 124), but
- adding a parity check bit [i] would yield OA(2243, 250, S2, 125), but
- the (dual) Plotkin bound shows that M ≥ 904 625697 166532 776746 648320 380374 280103 671755 200316 906558 262375 061821 325312 / 63 > 2243 [i]
- adding a parity check bit [i] would yield OA(2243, 250, S2, 125), but
- extracting embedded orthogonal array [i] would yield OA(2242, 249, S2, 124), but