Balancing cyclic R-ary Gray codes

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1 Balacig cyclic R-ary Gray codes Mary Flahive Departmet of Mathematics Orego State Uiversity, Corvallis, OR 97331, USA Bella Bose School of Electrical Egieerig ad Computer Sciece Orego State Uiversity, Corvallis, OR 97331, USA Submitted: Sep 26, 2006; Accepted: Apr 17, 2007; Published: Apr 27, 2007 Mathematics Subject Classificatio: 05A99,05C45,68R10 Abstract New cyclic -digit Gray codes are costructed over {0, 1,..., R 1} for all R 3, 2. These codes have the property that the distributio of the digit chages (trasitio couts) is close to uiform: For each 2, every trasitio cout is withi R 1 of the average R /, ad for the 2-digit codes every trasitio cout is either R 2 /2 or R 2 /2. 1 Itroductio For R, 2, a -digit R-ary Gray code is a sequece i which each -digit strig with digits from the set {0, 1,..., R 1} occurs exactly oce ad ay two cosecutive strigs differ i oly oe digit ad the differece equals ±1; that is, the Lee distace equals 1, where the Lee distace betwee two -strigs is defied to be d L (a 1... a, b 1... b ) = mi{ a j b j, R a j b j }. j=1 Whe the Lee distace betwee the last ad first strigs also equals 1, the code is called cyclic. For istace, 10, 11, 01, 00 ad 20, 21, 22, 12, 02, 01, 11, 10, 00 (1) are cyclic 2-digit R-ary Gray codes, with R = 2 ad R = 3, respectively. the electroic joural of combiatorics 14 (2007), #R31 1

2 It is helpful to cosider a cyclic Gray code as a Hamiltoia cycle i the R-ary - cube, the graph whose vertices are all -strigs with digits from the set {0, 1,..., R 1} i which two vertices are adjacet if they differ i oly oe digit by ±1. Our usual represetatio of this graph will have the vertices arraged i a R 1 R array, where the first 1 digits of a vertex -strig are give by its row label ad the last digit by its colum label. Figure 1 has the Hamiltoia cycles for the cyclic Gray codes i (1). Figure 1: cyclic 2-digit R-ary Gray codes for R = 2, 3, 4. The trasitio sequece of a cyclic Gray code records the successive digit chages i the code, begiig with the chage from the first to the secod strig. For example, the trasitio sequeces for the codes i (1) are: 2, 1, 2, 1 ad 2, 2, 1, 1, 2, 1, 2, 1, 1. (2) For each 1 j, the trasitio cout T C(j) of the digit j is defied to be the umber of times j occurs as a trasitio digit i the code. For each, R there are may Gray codes, ad oe favorable characteristic for applicatios is a relative uiformity i the distributio of trasitio couts. Whe all trasitio couts are equal (this commo value must ecessarily equal R / ad must divide R ), the code is called totallybalaced or uiform. The biary ad quaterary codes i Figure 1 are totally-balaced. I [7, 10, 11] totally-balaced cyclic -digit biary codes are costructed for every which is a power of 2, but there is o published costructio of families of totally-balaced codes for arbitrary R. Oe of our results yields totally-balaced quaterary codes. (Refer to Corollary 1.) Whe R is ot divisible by, totally-balaced cyclic codes caot exist ad it is reasoable to ask for codes i which every trasitio cout is either R / or R /, a pheomeo which we will call well-balaced. Whe R is eve, the cyclicess forces every trasitio cout to be eve ad so (ulike the cyclic terary code give above i (1)) for eve R every well-balaced cyclic R-ary Gray code must be totally-balaced. I [2], the authors asked if a well-balaced (o-cyclic) -digit biary Gray code exists for every 2. Although this cotiues to be a ope questio, i [9, Chapter 3] I. the electroic joural of combiatorics 14 (2007), #R31 2

3 N. Suparta has obtaied further evidece supportig the cojecture. I Sectio 2 we costruct well-balaced 2-digit R-ary Gray codes for every R 2. The codes deped o the eve/odd parity of R, ad are produced by a atural iductive process i which well-balaced cyclic 2-digit (R + 2)-ary Gray codes are costructed from the R-ary oes. A cyclic -digit biary Gray code satisfyig T C(i) T C(j) 2 for every pair i, j, is referred to as balaced. I [2], G. S. Bhat ad C. Savage costructed balaced cyclic -digit (biary) Gray codes for every 2 whe they completed some details which remaied from the costructio of J. P. Robiso ad M. Coh [7]. I a commet added to [2], F. Ruskey oted that T. Bakos [1, pp ] had much earlier costructed balaced cyclic biary codes; more recetly I. N. Suparta [9, Sectio 3.2] has developed a elegat variat of these costructios. I each, -digit cyclic biary codes are costructed iductively, usig four ( 2)-digit biary codes. No-biary codes have bee cosidered, for example, i [3, 4, 5, 6, 8, 9], but the questio of costructig fairly uiform cyclic R-ary Gray codes for a geeral radix R > 2 has received far less attetio tha the biary case. I Sectio 3 we give a simple iductive costructio which we later use i Sectio 4 to obtai early-balaced codes; that is, codes whose trasitio couts satisfy T C(j) R < ρ for all 1 j, where ρ equals R 1 or R 2, whichever is eve. The earlier costructios of well-balaced biary codes relied o delicate combiatorial argumets, whereas our use of R / as a referece poit has the advatage of allowig simpler proofs which ivolve oly balacig iequalities. 2 Well-balaced cyclic 2-digit R-ary Codes Theorem 1. Let R 2. If there exists a cyclic 2-digit R-ary Gray code with trasitio couts R 2 /2 +a ad R 2 /2 +b for some costats a, b, the there exists a cyclic 2-digit (R + 2)-ary Gray code with trasitio couts (R + 2) 2 /2 + a ad (R + 2) 2 /2 } + b. Proof. The give cyclic R-ary code ca be ormalized so that it begis with R-1 0 ad eds with the strig 0 0. (This ca be doe because at some poit there must be cosecutive terms of the form c d ; c ± 1 d. The required form is the obtaied by traslatig ad reflectig if ecessary so that the last term is 0 0 ad the first term is R-1 0. Sice the distributio of trasitio couts remais the same uder these operatios, we may assume the origial cyclic code has this form.) Therefore, the code correspods to a Hamiltoia path from R-1 0 to 0 0 i the R-ary square to which the edge from 0 0 to R-1 0 ca be added to form a Hamiltoia cycle. Labelig the rows of the (R + 2)-ary square as R+1 R ad the colums as R+1, we first costruct a path from the vertex R+1 0 to R-1 0 by begiig with the the electroic joural of combiatorics 14 (2007), #R31 3

4 vertex R+1 0, proceedig horizotally through all vertices of the form R+1 i, from R+1 0 to R+1 R+1, ad the vertically dow to 0 R+1. From there we travel to 0 R ad cotiue by proceedig vertically upward to R R, travelig horizotally over to R 0 ad the edig with the edge to R-1 0. This iitial path covers all vertices i which at least oe compoet is either R or R+1. Its trasitio sequece is: R + 1 twos, followed by R + 1 oes, oe 2, ad the R oes, R twos ad oe 1, for a total of 2R + 2 copies of each of 1 ad 2. Attachig the give Hamiltoia path from R-1 0 to 0 0 to the ed of this iitial segmet results i a Hamiltoia path i the (R + 2)-ary square, ad the additio of the edge from 0 0 to R+1 0 results i a Hamiltoia cycle. (The quaterary code i Figure 1 was obtaied i this way from the biary code, ad the diagram i Figure 2 illustrates the geeral costructio.) Sice Figure 2: Costructio of the iitial segmet. R 2 /2 + 2R + 2 = (R + 2) 2 /2 ad R 2 /2 + 2R + 2 = (R + 2) 2 /2, the code produced is a cyclic 2-digit (R + 2)-ary Gray code with the required trasitio couts. Theorem 2. For ay R 2 there exists a cyclic 2-digit R-ary Gray code which is wellbalaced, ad so for eve R the code is totally-balaced. Proof. Cosider the statemet: For every 1 there exist well-balaced cyclic 2-digit R-ary Gray codes for R = 2 ad R = The codes give i (1) satisfy this statemet for = 1. Applicatio of Theorem 1 with a = b = 0 therefore yields the result by iductio o. the electroic joural of combiatorics 14 (2007), #R31 4

5 3 Costructig ( + 1)-digit codes from -digit codes Oe simple way to costruct a Hamiltoia path o a L R rectagular grid is to traverse successive vertices across rows; that is, 1 0, 1 1,..., 1 R 1 ; 2 R 1,..., 2 1, 2 0 etc., (3) as illustrated i Figure 3. Notice that the termial vertex is either L 1 or L R, accordig Figure 3: Our basic Hamiltoia path o the 4 5 grid. to whether L is eve or odd. Our costructio of early-balaced cyclic codes is based o a modificatio of this simple idea. The costructio ivolves partitioig the row idices of a R-ary ( + 1)- cube ito blocks. Specifically, the rows are first idexed by a fixed cyclic -digit R-ary Gray code, a 1, a 2,..., a R, ad the are partitioed ito L oempty blocks of cosecutive elemets, say B 1 := a 1,..., a i1, B 2 := a i1 +1,..., a i2,..., B L := a il 1 +1,..., a R. This idea of usig partitios is at least implicit i other published costructios, amog them [2, 6, 7, 9, 10]. The digit chages from a ik to a ik+1 will be called the coectig digits of the partitio, ad the umber of times ay digit occurs as a coectig digit will be referred to as its coectig multiplicity. For otatioal coveiece, we assume the code is i a stadard form i which the coordiates of the -strigs have bee permuted if ecessary so that 1 is the L-th coectig digit, the trasitio digit from a R to a 1. We use this partitio to costruct a Hamiltoia path i the R-ary ( + 1)-cube i the followig way: First of all, the Hamiltoia path will respect the partitio B 1... B L ; that is, for every colum idex k ad every block B j, the path must traverse the vertices a ij +1 k, a ij +2 k,..., a ij+1 k cosecutively either i that directio or i the reverse directio. This allows us to picture the ( + 1)-cube as a L R array i which the rows are idexed by the blocks B 1,..., B L. the electroic joural of combiatorics 14 (2007), #R31 5

6 Figure 4: Terary Hamiltoia paths for three ad four blocks. If R is odd, the costructio give i (3) yields a Hamiltoia path o the R-ary ( + 1)-cube (refer to Figure 4), ad whe the umber L of partitio-blocks is eve umber, the additio of the edge from a R 0 to a 1 0 results i a Hamiltoia cycle. Whe R is eve, the path ca be adjusted to get a Hamiltoia cycle o the R-ary ( + 1)-cube i the followig way: Withi the give R-ary ( + 1)-cube (with its rows labeled by the cyclic Gray code which has bee partitioed by B 1... B L ad its colums labeled R-1), cosider the R (R 1) grid cosistig of the vertices whose last digit is ozero. Sice R 1 is odd, the Hamiltoia path give i (3) ca be costructed o the blocks of this grid. Its iitial vertex is a 1 1 ad the termial vertex is either a R R-1 or a R 1, depedig o whether L is odd or eve. I either evet, the termial vertex is adjacet to a R 0. A Hamiltoia cycle o the R-ary ( + 1)-cube is obtaied by appedig this edge to the iitial path, followig through all vertices with zero secod digit i the order: a R 0, a R 1 0,..., a 1 0, ad the edig with the edge to a 1 1. This is pictured for R = 4 i Figure 5. Regardless of whether R is eve or odd, the costructio gives a cyclic ( + 1)-digit R-ary Gray code (provided the umber of partitio-blocks is eve whe R is odd). I what follows this will be referred to as the code iduced by the partitio B 1... B L. By arragig the vertices of the (biary) -cube i a grid whose rows are idexed by a ( 2)-cube ad its colums by , most kow -digit biary codes ca be see to have this kid of partitioig behavior. Whe viewed from this perspective of partitios, the combiatorial argumet for the biary costructios i [2, 7] required some restrictios o the usable partitios; for istace, at least two blocks ca have oly oe elemet. The followig result gives the distributio of trasitio couts i codes iduced by a partitio. the electroic joural of combiatorics 14 (2007), #R31 6

7 Figure 5: Costructio of cyclic 4-ary codes. Theorem 3. Let R 3 be a iteger, ad set ρ equal to either R 2 or R 1, whichever is eve. Let a 1, a 2,..., a R be ay cyclic -digit R-ary Gray code with trasitio couts T 1,..., T (where two digit positios have bee trasposed if ecessary so that the digit chage from a R to a 1 is i the first digit). Let B 1... B L be ay partitio of the code, ad k j be the coectig multiplicity of the digit j. Whe R is odd ad L is eve, the j-th trasitio cout of the iduced cyclic ( + 1)-digit Gray code is { R T j k j ρ if j T C(j) = L ρ if j = + 1. (4) Whe R is eve the trasitio couts of the iduced cyclic ( + 1)-digit Gray code are R T j k j ρ 2 if j = 1 T C(j) = R T j k j ρ if 1 < j. (5) L ρ + 2 if j = + 1 Proof. First we cosider the cotributio from the iitial Hamiltoia path. Settig N := ρ+1 (which is odd), the Hamiltoia path o the R N grid has (N-1)L horizotal lies, ad so this iitial part of the process of formig the Hamiltoia cycle accumulates (N-1)L chages i the digit + 1. Every edge withi the partitio-blocks is traversed i every colum, the edge correspodig to the coectig digit from a R to a 1 ever occurs, ad the edge correspodig to every other coectig digit occurs exactly oce. Therefore, for every 1 < j the digit j chages N (T j k j ) + k j = N T j (N 1)k j times i this iitial segmet, ad there is oe fewer chage for j=1. Whe R is odd, the cycle is completed by addig oe edge which correspods to a chage i the first digit. Sice N = R holds i this case, (4) is obtaied. the electroic joural of combiatorics 14 (2007), #R31 7

8 For eve R, two horizotal edges are added to complete the cycle, givig T C( + 1) = (N 1)L + 2 = L ρ + 2. As for additioal vertical edges, every edge i the origial code except for the oe correspodig to trasitioig from a R to a 1 is added oce. Sice the trasitio from a R to a 1 is a chage i the first digit, this gives ad for all 2 j, This proves (5). T C(1) = (R 1) T 1 (R 2)k T 1 1 = R T 1 k 1 ρ 2, T C(j) = (R 1) T j (R 2)k j + T j = R T j k j ρ. We ed this sectio with the costructio of a cyclic 3-digit terary code iduced from a partitio of the 2-digit terary code i (1). Sice R is odd, the umber of partitioblocks must be eve. Also, the digit 1 occurs more frequetly as a trasitio digit i the give 2-digit code tha 2 (refer to (2)) ad so a preferred partitio would have 1 occurrig at least as ofte as a coectig digit as 2 does. Cosider ay 4-block partitio i which 2 occurs oly oce as a coectig digit. (For istace, B 1 := {20, 21}, B 2 := {22}, B 3 := {12, 02, 01}, B 4 := {11, 10, 00} is such a partitio, sice the coectig digits are 2,1,1,1.) From (4), the iduced code has T C(1) = 9 ; T C(2) = 10 ; T C(3) = 8. It is ot totally balaced but is early-balaced sice T C(j) R / 1 holds for all j. 4 Nearly-balaced cyclic R-ary Gray Codes Lemma 1. Let R 3 be a iteger. If T is ay trasitio cout of either a well-balaced 2-digit R-ary code or a early-balaced -digit R-ary code for 3, the there exists a uique iteger k such that 0 < k < T ad 0 R T k ρ R < ρ. (6) Proof. For k := (R T R+1 )/ρ, we see that (6) is satisfied. Rearragig (6), +1 ad so 0 < k < T ca be proved by showig R T R ρ < k ρ R T R+1 + 1, R T R+1 R+1 ρ > 0 ad R T < T ρ ; the electroic joural of combiatorics 14 (2007), #R31 8

9 that is, R T R+1 R+1 ρ > 0 ad T (ρ R) > 0. (7) Cosiderig the case of a well-balaced 2-digit code, every trasitio cout T satisfies givig R T R R T R3 3 ρ R3 9R for R 3, ad sice R ρ equals 2 or 1,, > 0 T (ρ R) + R3 3 R2 + 1 ( 2) + R3 2 3 = R2 (R 3) 3 3 which is positive for all R 4. Also, for R = 3, T (ρ R) + R 3 /3 = T provig (7) for well-balaced 2-digit codes. It remais to cosider the trasitio couts of a early-balaced -digit code with 3; that is, T R < ρ, where we recall that ρ equals either R 1 or R 2, whichever is eve. The first iequality i (7) ca be easily checked for R = = 3 sice ρ = 2 ad the trasitio couts satisfy 8 T 10. Defiig a := R +1 /( + 1), we observe that for R, 3 a +1 = R ( 1 2 ) > 1, a + 2 ad so a a 3 = R 4 /12 for all 3. Sice T is a trasitio cout of a early-balaced -digit code, T R / < ρ ; T > R ρ ad ρ R 1 imply R T R ρ > R ( R ρ ) R ρ = a (R + 1)ρ a 3 (R + 1)(R 1) = R2 (R 2 12) which is positive for R 4. The first iequality has already bee proved for R = = 3. For R = 3 ad 4, we observe that a (R + 1)ρ a = > 0. This proves the first iequality i (7) for all R, 3. the electroic joural of combiatorics 14 (2007), #R31 9

10 For the secod iequality: T < ρ + R ad ρ < R give ( T (ρ R) + R > ρ + R = ρ 2 + )(ρ R) + R ( R R ) ρ R+1 ( + 1), a strictly icreasig fuctio of ρ R 2. Evaluatio at ρ = R 2 gives ( R (R 2) 2 R(R 2) + R ) = 2(R 2) + b, ( R where b := R ). We ote that b +1 b = R + 2 ( + 1)R 2( + 2) R 2( + 1) where the restrictios R 3 ad 3 imply each of the two idetified factors is greater tha 1. Therefore, b b 3 ad so 2(R 2) + b 2(R 2) + b 3 = 3R4 8R 3 24R + 48, 12 which is positive for all R 3. This proves (7) ad so the lemma. Theorem 4. For every iteger R 3 ad every 2 there exists a early-balaced cyclic -digit R-ary Gray code. Proof. Let R 3 be a fixed iteger. Sice every well-balaced code is early-balaced, Theorem 2 serves as the base case for a proof by iductio o 2. Let T 1,..., T be the trasitio couts of ay 2-digit well-balaced code or ay earlybalaced -digit code for 3. We will prove there is a partitio of this code for which the iduced cyclic ( + 1)-digit code is early-balaced. From Lemma 1 it follows that for each j = 1,..., there exists a uique iteger k j with 1 k j < T j such that satisfies S j := R T j k j ρ R (8) 0 S j < ρ. (9) Permute the coordiates of the -digit code if ecessary to obtai ρ > S 1 S 2... S 0. (10) Sice each 1 k j < T j, the digit j occurs more tha k j times as a trasitio digit i the code ad so it ca be partitioed ito L := j=1 k j blocks i such a way that each digit j occurs exactly k j times as a coectig digit. Usig (8) ad j=1 T j = R, S j = R ρ L. (11) j=1 the electroic joural of combiatorics 14 (2007), #R31 10

11 Let M := j=1 S j/ρ. The (9) implies 0 M <. (12) Also, by defiitio of M, We also ote that if the from (11) ad (9), 0 j=1 S j M ρ < ρ holds ad so (11) gives 0 R+1 ρ (L + M) < ρ. (13) + 1 S M+1 =... = S = 0, R +1 ρ (L + M) = + 1 j=1 S j M ρ = M S j M ρ < M ρ M ρ = 0, j=1 cotrary to (13). From (9) we therefore have 0 < S j < ρ for all j M + 1 ad 0 S j < ρ for all j > M + 1. (14) We distiguish two cases, accordig to whether R is odd or eve. CASE 1: R 3 is odd. We will prove there exists a refiemet of the above partitio for which the iduced ( + 1)-digit code is cyclic ad early-balaced. Sice R is odd, the umber L of partitio-blocks must be eve for the code iduced by the partitio to be cyclic. Our refiemet will have L partitio-blocks where L equals either L + M or L + M + 1, whichever is eve, ad so D := L L coectig digits will be added. Sice D equals M or M + 1 ad 1 k j < T j, by (12) 0 D ad each of the first D digits ca be chose oe more time as a coectig digit, givig L partitio-blocks with multiplicities k 1,..., k satisfyig k j = k j + 1 if j D, k j = k j if j > D. Accordig to (4), the first trasitio couts of the code iduced by this ew partitio have { T C(j) R = S j ρ if j D S j if D < j, which by (14) satisfy the early-balaced coditio. Also, the last trasitio cout is T C( + 1) = L ρ, ad (13) implies R +1 R+1 ρ < (L + M) ρ T C( + 1) (L + M + 1) ρ ρ. Therefore, the code is early-balaced uless (L + M) ρ = R +1 /( + 1), the electroic joural of combiatorics 14 (2007), #R31 11

12 which caot happe sice ρ is eve ad R is odd. This completes the proof for odd R. CASE 2: R 4 is eve. We refie the partitio by addig oe more coectig digit for each j = 2, 3,..., M + 1. This iduces a code with L + M partitio-blocks whose first trasitio digits satisfy T C(j) R = S 1 2 if j = 1 S j ρ if 1 < j M + 1 if M + 1 < j S j As i Case 1, (14) implies the early-balaced coditio for all T C(j) with j. Also, sice T C( + 1) = (L + M) ρ + 2, from (13) we have ρ 2 R+1 T C( + 1) < ρ 2 < ρ, + 1 ad so the code is early-balaced uless ρ = 2 (ad so R = 4) ad T C( + 1) = R For this special case, cosider ay modificatio of the last partitio which has oe less occurrece of 1 as a coectig digit. This chages oly the first ad last trasitio couts to T C(1) + 2 = S 1 + R+1 ad T C( + 1) ρ = T C( + 1) 2 = R , respectively, resultig i a early-balaced code. We ext apply this theorem to the case whe divides R. Corollary 1. Let R 3 ad 2 be itegers. If divides R, the there exists a cyclic -digit R-ary Gray code whose trasitio couts T C(1),..., T C() satisfy T C(j) R ρ 1 for all 1 j. (15) I particular, whe is a power of 2 there exist totally-balaced cyclic -digit quaterary Gray codes. Proof. By hypothesis, T C(j) R is a iteger, ad so the strict iequality i the defiitio of early-balaced implies (15). Whe R = 4 ad = 2 k, divides R ad so every trasitio cout of a earlybalaced cyclic -digit quaterary code is withi ρ 1 = 1 of the eve iteger 4 /. Sice the code is cyclic ad R is eve, every trasitio cout must be eve ad so must equal 4 /. We ed with a applicatio to biary codes.. the electroic joural of combiatorics 14 (2007), #R31 12

13 Corollary 2. For each 2 there exists a cyclic 2-digit biary Gray code whose T C(2i 1)+T C(2i) trasitio couts T C(1),... T C(2) are such that every average of the form 2 equals either 2 2 /(2) or 2 2 /(2). Proof. Cosider the followig mappig of elemets of Z 4 to 2-digit biary strigs: 0 00 ; 1 01 ; 2 11 ; This mappig exteds to a mappig of quaterary -strigs to biary 2-strigs uder which Lee distace is preserved. Therefore, ay cyclic -digit quaterary Gray code is mapped to a cyclic 2-digit biary Gray code, ad the trasitio couts T C(1),... T C(2) of the biary code have the property that T C(2i 1)+T C(2i) equals the trasitio cout of the digit i i the 4-ary code for every i = 1,.... Usig ay early-balaced -digit 4-ary Gray code, we therefore have T C(2i 1) + T C(2i) = T C(2i 1) + T C(2i) < 1, from the defiitio of early-balaced. Sice every trasitio cout of a cyclic biary Gray code is eve, each such average is a iteger ad the last iequality implies it must equal either 2 2 /(2) or 2 2 /(2), as claimed. Ackowledgmet: The authors are grateful to the two referees for their helpful suggestios, especially for the simplificatio of the proof of Theorem 4. Refereces [1] A. Ádám. Truth fuctios ad the problem of their realizatio by two-termial graphs. Akadémiai Kiadó, Budapest, pages [2] Girish S. Bhat ad Carla D. Savage. Balaced Gray codes. Electroic Joural of Combiatorics, 3(1):Research Paper 25, approx. 11 pp. (electroic), [3] B. Bose ad B. Broeg. Lee distace Gray codes. I Proceedigs of the Iteratioal Symposium o Iformatio Theory, [4] B. Bose, B. Broeg, Y. Kwo, ad Y. Ashir. Lee distace ad topological properties of k-ary -cubes. IEEE Trasactios o Computers, 44: , [5] Marti Coh. Affie m-ary Gray codes. Iformatio ad Cotrol, 6:70 78, [6] Doald E. Kuth. The Art of Computer Programmig, volume 4, Fascicle 2. Addiso- Wesley, [7] Joh P. Robiso ad Marti Coh. Coutig sequeces. IEEE Trasactios o Computers, 30(1):17 23, [8] Bhu Dev Sharma ad Ravider Kumar Khaa. O m-ary Gray codes. Iform. Sci., 15(1):31 43, the electroic joural of combiatorics 14 (2007), #R31 13

14 [9] I Negah Suparta. Coutig sequeces, Gray codes ad Lexicodes. PhD thesis, Delft Uiversity of Techology, [10] A. J. va Zate ad I. N. Suparta. Totally balaced ad expoetially balaced Gray codes. Discrete Aalysis ad Operatios Research, Ser. I, 11:81 98, [11] David G. Wager ad Julia West. Costructio of uiform Gray codes. Cogr. Numer., 80: , the electroic joural of combiatorics 14 (2007), #R31 14

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018 CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical