MinT - Best Known (67−18, 67, s)-Nets in Base 8

Best Known (67−18, 67, s)-Nets in Base 8

(67−18, 67, 457)-Net over F₈ — Constructive and digital

Digital (49, 67, 457)-net over F₈, using

8¹ times duplication [i] based on digital (48, 66, 457)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁶⁶, 457, F₈, 18, 18) (dual of [(457, 18), 8160, 19]-NRT-code), using
  - OA 9-folding and stacking [i] based on linear OA(8⁶⁶, 4113, F₈, 18) (dual of [4113, 4047, 19]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁶⁶, 4114, F₈, 18) (dual of [4114, 4048, 19]-code), using
      - constructionÂ XX applied to C_e(17) âŠ‚ C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(8⁶¹, 4096, F₈, 18) (dual of [4096, 4035, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(8⁴⁹, 4096, F₈, 14) (dual of [4096, 4047, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(8⁴⁵, 4096, F₈, 13) (dual of [4096, 4051, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(8⁴, 17, F₈, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,8)), using
        discarding factors / shortening the dual code based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
        linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
        dual of repetition code with length 1 [i]

(67−18, 67, 644)-Net in Base 8 — Constructive

(49, 67, 644)-net in base 8, using

8¹ times duplication [i] based on (48, 66, 644)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 18, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 9, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. (30, 48, 514)-net in base 8, using
    - base change [i] based on digital (18, 36, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 18, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ 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]

(67−18, 67, 4242)-Net over F₈ — Digital

Digital (49, 67, 4242)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁷, 4242, F₈, 18) (dual of [4242, 4175, 19]-code), using
- 136 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 86 times 0) [i] based on linear OA(8⁶¹, 4100, F₈, 18) (dual of [4100, 4039, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(8⁶¹, 4096, F₈, 18) (dual of [4096, 4035, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(67−18, 67, 3130804)-Net in Base 8 — Upper bound on s

There is no (49, 67, 3130805)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 213883 239415 242831 475824 646999 593725 214421 654062 628644 631528 > 8⁶⁷ [i]

Best Known (67−18, 67, s)-Nets in Base 8

(67−18, 67, 457)-Net over F8 — Constructive and digital

(67−18, 67, 644)-Net in Base 8 — Constructive

(67−18, 67, 4242)-Net over F8 — Digital

(67−18, 67, 3130804)-Net in Base 8 — Upper bound on s

(67−18, 67, 457)-Net over F₈ — Constructive and digital

(67−18, 67, 4242)-Net over F₈ — Digital