1956 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS 139

CorrespondenceUnit-Distance Binary-Decimal one of the ten utilized digit locations, and the

K =ABC D progression 0-1-2 etc. can proceed in eitherCommutation *=s o o o o direction along the trace oni the map. ThusCommutation* I2 0 0o 1 a total of 96XlOX2, or 1920, unit-distance

In a recent letter to the editor' Bernard 3 0 o 1 o Lippel codes exist. It can be shown by ex-4 0 1 1 0 haustion of allowable transitional combina-

Lippel describes a decimal code for analog- 5 I I 1 0to-digital conversion which has the property 6 - 1 1 1 1 tioris that there are no other four-bit unit-7 1 1 01 itneLpe dcmlcdsthat only two commutator tracks (zones) 8 1 1 o distance Lippel decimal codes.are required for each decimal-digit decade 9 1 0 0 0 Logical detenting will still be requiredbeing converted from shaft position into between decades, if conventional rather thandigital form. For brevity, we shall designate reflected decimal output is desired, to insurecodes having this property "Lippel codes." K, = A, B, C, Di that in a multiple-digit transition (e.g., from

Three bits of the Lippel code representa- 0 0 0- 29 to 30) the high-order digit transitioii (2

tion of each decimal digit are obtained from i - o o 1 to 3) is commutated first, but that the con-a single track (zone), which has its brushes 2 = 0 0 1 0 verter output does not reflect this change3 0 0 11 itltetm htth oe-re iidisplaced allgularly so that they contact the 4 - o 1 1 o until the time that the lower-order digitsame track at differeiit points. A fourth 5 - 00o transition (9 to 0) occurs. If reflected deci-brush is used on a separate track to read the 7 1 0 1 o mal output can be used, nO logical detentingfourth bit of the coded representation. 8 - o 1 is needed, and only four brushes per decade

The specific code that Lippel describes is are required.not of unit distance.' Hence, to avoid tran- conersio fa unit-distace ippelsition errors within the decade Lippel sug- code to a standard weighted-count code is ingests using logical detenting, i.e., reading of K2 = A2 B2 C2 D2 general straightforward. For example, codegests ~usn loia deetig i.. redn1 of K=ABCD of Fig. 1 converts to the stand-the zone transitions in definite known se- 0 = 0 0 0 0 Kquence, and provision of a converter output I 0 o 0 1 ard 5-2-2-1 code, shown as K1=A2B1CID1 in

quence, andprovision ~~~~~~~~2= 0 0 1 0 Fig. 1, as follows:only when the last relevant zone transitioni 3 = 0 0 1 1occurs. Because of the logical detent require- 5 _0 1 0 1ment Lippel needs as many as eight brushes 6 = 0 1 1 0per decimal decade. If a unit-distance Lippel 7 _ 1 1 1 AiA8 I0 0 0code were used, the number of brushes re- 9 = 1 0 0 1quired per decade could be reduced from B1-ABvABeightdtopfourorde fve.coulipel implies i i Fig, 1-A specific unit-distance Lippel code, K;eight to four or five. Lippel implies in his the standard 5-2-2-1 code, K,; andletter that unit distance Lippel codes are standard 8-4-2-1 code, K2. Cl ACvACnot known.

The code designiated in Fig. 1 as D -BCDvBCDvBDvBCDK=ABCD is one of the family of unit-dis-tance Lippel codes. (Inicidentally, K is ofunit-distance but is not a reflected code.) CD

The family of uiiit-distance Lippel codes AB" 000wheren ist"if,"and dV oily if," thebar indicatescan be pictured on Veitch-Karnaugh B ' "not," aiid V means "or, . . . or both." TheMaps3-5 as indicated in Fig. 2. O'Brien has 0 0 X -)( X _ - - expression for Di is the triple-exclusive-or,recently studied reflected codes using 0 _Xanv onie or all three," which cani be realizedVeitch-Karnaugh Maps.6 by a relay symmetrical circuit.

Any of the 96 suibfamilies of codes pic- X-- -X- - a X- -X-- Conversion of the code K=ABCD intottired in Fig. 2 cani hav-e decimal 0 in any the ordinary 8-4-2-1 code, shown in Fig. 1

L ~~~~as K2=A2B2C2D2, is as follows:

* Received by tlle PGEC, jtily 16, 1956. The work ETC, BY SHIFTS,reported here was performiied while the author was at -XG E --XRATE AC-, ABurroughs Research Center, Paoli, Pa. A _2 A CD

1 B. Lippel, "A decimal code for analog-to-digital i 16 SUBFAMILIESconversion," IRE TRANS., vol. EC-4, pp. 158-159; IDecember, 1955. -*EAHBYHFSB2 BCvD

2 In a code of "unit distance," code words repre- EAvXGENERATESsenting consecutive numbers differ only in one digit GENERATESposition, and in that position the digits differ only by 16 SUBFAMILIES C2 AD v SCone. For example, the code "K" of Fig. 1 is of unitdistance. A 'reflected" or "cyclic" code is a special / tcase of a unit-distance code, in which the sequence of i I Ds AB v (A vB)(CD vCD).code words follows a definite pattern with the mth > -X_digit from the right changing by one if and only if the -X.(m-I)th digit is at its lowest or highest value, andwith each digit changing successively from its lowest . b y-to highest value by unit steps, then back from highest .This coversion is less conplex than con-to lowest value by unit steps. For example, the follow- A --X+ming numbers form a reflected decimal sequence: 01, a versio nof the codes recommended hy02, 03, 04, 05, 06, 07, 05, 09, 19, 18, 17,** *,12, 11, O-.'Br e 6n10, 20, 21, 22, *--. A recently published paper on re.reflected codes is I. Flores, "Reflected number sys- X_Further examinationl of unit-distance

tems,"IwEvRANhS uAlchEr-t5PPthodfo; sipiifyng §§ | xLippel codes, using Veitch-Karnaugh Maps,truth fulnctions," Proc. ACM, Pittsburgh Meeting. |-tXl ll - | | |X mgtwell reveal spcfccodes with simplerpp. 127-133, May 2-3, 1952. A--- li Xl lil code-conlversion rules; an exhaustive searchcombinational logic circuits," Trans. AIEE, Communz. II I|| tt;| || hsntytbe ae

KH11j mih specificIIi5S. H. Caldwell, "The recognition and identifica-

I8t^I II4 __

tion of symmetric switching functions," Trans. AIEE, 1, bHOWARD E. TOMPKINSCommun. anzd Elec., vol. 73, pt. I, pp. 142-147; May, |__r ii I -t Moore School of Elec. Enlgrg.,

6 J. A. 0'Brien, "Cyclic decimal codes for analog- Un1iv. of Pennlsylvanlia,to-digital converters," Trans. AJEE, Commun. a,,d Fig. 2-Topology of unit-distance Lippel codes onPhldpia4P.Elec., vol. 75. pt. I, pp. 120-122; May, 1956. Veitch-Karnaugh Maps. hadpl 4P.