218 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Table IV — Region of the n-k Plane for Which it is Known 



THAT [n, fc)-ALPHABETS CaNNOT Be EXCELLED 



k 



30 



29 



28 • • • 



27 .... 



26 



25 



24 



23 



22 



21 



20 



19 



18 



17 



16 



15 



14 



13 



12 



11 



10 



9 



8 



7 



6 



5 



4 



3 



2 



1 



\ 



1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 n 



This can be done by a \'ariety of special methods which considerably 

 reduce the obvious labor of making such an array. A set of best »S's along 

 with their parity check symbols is given in Table V. 



A maximum likelihood detector for the (10, 5)-alphabet in question 

 forms from each received sequence 6162 • • • &10 the parity check symbol 

 C1C2C3C4C5 where 



Ci = h 4- ^h 4- ^3 + Ih + ^5 



C2 = 67 -(- 6i -]- h-i + hi \- Ih 



Cs = &8 + ^^1 + h -j- Ih + ^5 



Ci = bg + hi 4- h-i -i- h-.i -(- hi 



C5 = />in + hi + />, + h, 4- hi 4- 65 



According to Table V, if CiC-jCiAf'b contains less than three ones, the de- 

 tector should brint hih^kihih^ . The detector should piint (/m 4- 1)^2^3^4'':. 

 if the parity check sequence C1C2C3C4C5 is either 11111 oi- 11110; the dv- 



