Designs, Codes and Cryptography, 27, 145–156, 2002©C 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Existence of Perfect 4-Deletion-Correcting Codeswith Length Six

NABIL SHALABYDepartment of Mathematics, Memorial University of Newfoundland, St. John’s, NF, Canada A1C 5S7

JIANMIN WANGDepartment of Mathematics, Suzhou University, Suzhou 215006, China

JIANXING YIN∗† jxyin@suda.edu.cnDepartment of Mathematics, Suzhou University, Suzhou 215006, China

Communicated by: C. J. Colbourn, D. R. Stinson, G. H. J. van Rees

Abstract. By a T ∗(2, k, v)-code we mean a perfect 4-deletion-correcting code of length 6 over an alphabet ofsize v, which is capable of correcting any combination of up to 4 deletions and/or insertions of letters that occurin transmission of codewords. The third author (DCC Vol. 23, No. 1) presented a combinatorial construction forsuch codes and prove that a T ∗(2, 6, v)-code exists for all positive integers v �≡ 3 (mod 5), with 12 possibleexceptions of v. In this paper, the notion of a directed group divisible quasidesign is introduced and used to showthat a T ∗(2, 6, v)-code exists for all positive integers v ≡ 3 (mod 5), except possibly for v ∈ {173, 178, 203, 208}.The 12 missing cases for T ∗(2, 6, v)-codes with v �≡ 3 (mod 5) are also provided, thereby the existence problemfor T ∗(2, 6, v)-codes is almost complete.

Keywords: codes, deletion/insertion-correcting, designs

1. Introduction

Let Q be an alphabet of size v, or equivalently a v-set (of points), and Qk the set of allvectors (called words) of length k over Q. We say that a word x of length n is a subwordof a word y of length m (n ≤ m) if x can be obtained from y by deleting m − n letters. Forexample, (2,1,0,4) is a subword of (2,0,1,1,0,4,3).

A subset C ⊆ Qk is said to be a perfect (k − 2)-deletion-correcting code over Q if everyword of Q2 occurs as a subword in exactly one word of C . Following [3], we call such codea T ∗(2, k, v)-code. As usual, the words of C are referred to as codewords.

We remark that a T ∗(2, k, v)-code is capable of correcting any combination of up to(k − 2) deletions and/or insertions of letters that occur in transmission of codewords. This

∗Corresponding author.†Research supported in part by NSFC Grant 10071056.

146 SHALABY ET AL.

follows that no two words in a T ∗(2, k, v)-code can share a common subword of length 2or longer. For more details and related topics, the reader is referred to [3,6–7].

Motivated by its diverse applications in several fields such as speech processing andmacromolecular biology [5], Levenshtein [6] introduced the notion of a perfect insertionand/or deletion correcting-code. The existence of a T ∗(2, k, v)-code has been now com-pletely solved for 3 ≤ k ≤ 5 (see [3,6–7]). Recently, the third author [8] investigated theexistence of a T ∗(2, 6, v)-code and proved the following.

THEOREM 1.1. For all positive integers v �≡ 3 (mod 5), a T ∗(2, 6, v)-code exists, exceptpossibly for v ∈ {20, 21, 86, 87, 99, 100, 101, 115, 116, 117, 146, 147}.

Here we establish the existence result of a T ∗(2, 6, v)-code with v ≡ 3 (mod 5). Weshall introduce a new combinatorial configuration, which we call a directed group divisiblequasidesign (DGDQD), and use it to obtain the following theorem.

THEOREM 1.2. For all v ≡ 3 (mod 5), there exists a T ∗(2, 6, v)-code, except possibly forv ∈ {173, 178, 203, 208}.

The 12 missing cases for T ∗(2, 6, v)-codes in Theorem 1.1 are also provided in thispaper. This complements the earlier study by the third author. The combined results of thetwo papers can be summarized into the following theorem.

THEOREM 1.3. For all positive integers v, there exists a T ∗(2, 6, v)-code, except possiblyfor v ∈ {173, 178, 203, 208}.

We use [2] and [4] as our design theory references. A group divisible design (GDD) ofindex unity and block sizes from K is denoted by K -GDD. When K = {k}, it is simplydenoted by k-GDD. An ‘exponential’ notation is adopted to indicate the list of the groupsizes (called type) of a K -GDD. A transversal design (TD) with block size k and groupsize n is denoted by TD(k, n). The notation TD(k, n)–TD(k, m) stands for an incompletetransversal design (ITD), that is, a TD(k, n) with all blocks of a sub-TD(k, m) removed. Thenotation B(k, 1; v) stands for a balanced incomplete block design (BIBD) with parameters(v, k, 1).

Following [8], a T ∗(2, k, v)-code C1 over an alphabet U is referred to as a subcode ofa T ∗(2, k, v)-code C2 over an alphabet V if C1 ⊆ C2 and U ⊆ V . We adopt the notationT ∗

u (2, k, v)-code to denote a T ∗(2, k, v)-code that contains a T ∗(2, k, u)-code as its subcode.We admit u = 0. If the code is over an empty alphabet, then it is considered to be a subcodeof any code. The existence of a T ∗

u (2, k, v)-code implies the existence of a T ∗(2, k, v)-code,but the converse is not true unless u = 0.

We shall make use of the following three known results, which are all taken from [4].

LEMMA 1.4. A TD(6, n) exists for any integer n ≥ 5 except for n = 6 and possibly forn ∈ {10, 14, 18, 22}, and a TD(7, n) exists for any integer n ≥ 7 except possibly for n ∈{10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 54, 60, 62}.LEMMA 1.5. If (n, m) ∈ {(52, 2), (53, 3), (54, 4), (77, 2)}, then a TD(6, n)–TD(6, m)

exists.

PERFECT 4-DELETION-CORRECTING CODES 147

LEMMA 1.6. Let v be a positive integer. If v ≡ 1 or 6 (mod 15), then a B(6, 1; v) exists, ex-cept when v ∈ {16, 21, 36} and possibly when v ∈ {46, 51, 61, 81, 166, 226, 231, 256, 261,

286, 291, 316, 321, 346, 351, 376, 406, 411, 436, 441, 471, 496, 501, 526, 561, 591, 616,

646, 651, 676, 771, 796, 801}.

2. The Main Construction

Let v and k be positive integers. A transitively ordered k-tuple (a1, a2, . . . , ak) is definedto be the set {(ai , a j ) : 1 ≤ i < j ≤ k} consisting of k(k − 1)/2 ordered pairs. A directedgroup divisible design (DGDD) with block size k and index unity, denoted by k-DGDD, isa triple (X ,G,B) which satisfies the following properties:

(1) G is a partition of a set X (of points) into subsets called groups,

(2) B is a collection of transitively ordered k-tuples (blocks) of X , such that a group and ablock contain at most one common point,

(3) every ordered pair of points from distinct groups occurs in exactly one of blocks.

The group-type (type) of the DGDD is the multi-set {|G| : G ∈ G}. As with GDDs, weusually use exponential notation to describe the type of the DGDD. Using this notation, adirected balanced incomplete design (DBIBD), DB(k, 1; v), can be defined to be a k-DGDDof type 1v . An incomplete DBIBD (IDBIBD), IDB(k, 1; v, w), can be defined as a k-DGDDof type 1v−w w1, in which the group of size w is the hole.

For the existence of DGDDs, we record the following two known results.

LEMMA 2.1 [1]. The necessary and sufficient condition for the existence of a DB(6, 1; v)

is that v ≡ 1 or 6 (mod 15), v ≥ 6 and v �= 21.

LEMMA 2.2 [1,8]. There exist 6-DGDDs with the following types:

26, 46, 57, 59, 510, 512.

We remark that if a k-GDD exists, then so does a k-DGDD. The k-DGDD is obtained bywriting all blocks of the k-GDD twice—once in some order and the other in reverse order.

Now we introduce the notion of directed group divisible quasidesigns. A directed groupdivisible quasidesign, or a DGDQD, is a triple (X ,G,B) where X is a finite set (of points),G is a partition of X into subsets (called groups), and B is a collection of sequences (calledblocks) of length k over X with the following properties:

(1) every ordered pair of points from distinct groups occurs as a subsequence in exactlyone block;

(2) for any point x in all but one distinguished group, the pair (x, x) occurs as a subsequencein a unique block, while for any point y in the distinguished group the pair (y, y) doesnot occur in any block; and

(3) all pairs of distinct points from the same group do not occur together in any block.

148 SHALABY ET AL.

If the distinguished group has size d and all other groups have the same size g, then wecall the DGDQD a k-DGDQD of type gt d1 where t is the number of groups of size g.

According to the above definitions, adding blocks of the form (x, x, . . . , x) to a k-DGDDof type gt d1 yields a k-DGDQD of type gt d1 where x runs over all groups except for thegroup of size d . Thus the existence of a k-DGDD of type gt d1 implies the existence of ak-DGDQD of type gt d1. However, the converse is not true. For example, by the necessarycondition for the existence of DGDDs there does not exist a 6-DGDD of type 5621, but wedo have the following result.

LEMMA 2.3. There exists a 6-DGDQD of type 5621.

Proof. For this DGDQD, the point set X is taken to be Z30 plus two infinity points{∞1, ∞2} which make up the distinguished group of size 2. The groups of size 5 are{ j, j + 6, j + 12, j + 18, j + 24}, j = 0, 1, . . . , 5. Then the block set B consists of thefollowing 80 blocks:

(8, 0, 1, 16, 5, 3) (mod 30)

(1, 0, 14, 10, 21, ∞1) (+6, mod 30)

(∞1, 2, 1, 15, 11, 22) (+6, mod 30)

(3, 2, 16, 12, 23, ∞2) (+6, mod 30)

(∞2, 4, 3, 17, 13, 24) (+6, mod 30)

(5, 4, 4, 18, 14, 25) (+6, mod 30)

(6, 5, 19, 15, 15, 26) (+6, mod 30)

(5, 5, 5, 5, 5, ∞1) (+6, mod 30)

(∞1, 0, 0, 0, 0, 0) (+6, mod 30)

(1, 1, 1, 1, 1, ∞2) (+6, mod 30)

(∞2, 2, 2, 2, 2, 2) (+6, mod 30).

Here the notation (+6, mod 30) means that 6 (mod 30) should be successively added to theblock, generating five blocks.

The significance of DGDQDs as defined above is that their groups can be frequently filledin to give us T ∗(2, k, v)-codes. This approach leads to the following main construction,which is very effective in dealing with the problem under our consideration.

CONSTRUCTION 2.4. Let w be a non-negative integer. Suppose that there exist:

(1) a k-DGDQD of type gt d1;

(2) an IDB(k, 1, g + w, w); and

(3) a T ∗(2, k, d + w)-code.

Then there exists a T ∗u (2, k, v)-code, where v = gt + d + w and u = d + w.

Proof. Adjoin a set F of w infinite points to each group of the given DGDQD. Replaceall groups of size g plus F by IDB(k, 1; g + w, w)s so that F is their common hole of sizew . Replace the distinguished group of size d + w by a T ∗(2, k, d + w)-code. Note that

PERFECT 4-DELETION-CORRECTING CODES 149

the blocks of the DGDQDs and IDBIBDs can be taken as codewords. It can be easily checkedthat the above procedure produces a T ∗(2, k, gt +d +w)-code with a T ∗(2, k, d +w)-codeas its subcode.

Applying Construction 2.4 with k = 6, we have the following working lemma.

LEMMA 2.5. Suppose that there exists:

(1) a TD(7, n);

(2) an IDB(6, 1; 5n + w, w); and

(3) a T ∗(2, 6, 5s + 2 + w)-code with 0 ≤ s ≤ n − 1.

Then there exists a T ∗u (2, 6, v)-code where u = 5s + 2 + w and v = 30n + u.

Proof. Delete n − s − 1 points from a certain group in a TD(7, n) to yield a {6, 7}-GDDof type n6(s + 1)1. Choose a certain point, say x , in the group of size (s + 1) of theresulting GDD. Give weight 2 to the point x and weight 5 to the rest, and replace everyblock containing x by a 6-DGDQD of type 5621 and each of the remaining blocks by a6-DGDD of type 56 or 57 depending on its size. The ingredients used above all exist byLemmas 1.4, 2.2, and 2.3, where the 6-DGDD of type 56 follows from a TD(6, 5). Since theblocks containing x cover every point not in its group exactly once, the above procedureproduces a 6-DGDQD of type (5n)6(5s + 2)1. The conclusion then holds by Construction2.4.

3. T∗(2, 6, v)-Codes with Small Values of v

In this section, we present our constructions of T ∗(2, 6, v)-codes for certain small valuesof v. We shall use the following two basic constructions found in [8].

LEMMA 3.1. Let w be a non-negative integer. If there exist a k − DGDD of type{t1, t2, . . . , tr }, a T ∗(2, k, tr + w)-code and a T ∗

w (2, k, ti + w)-code for 1 ≤ i ≤ r − 1, thenthere exists a T ∗

u (2, k, t + w)-code where t = ∑1≤i≤r ti and u = tr + w.

LEMMA 3.2. Let w be a non-negative integer. If a TD(k, n)−TD(k, m), a T ∗m+w (2, k, n+w)-

code and a T ∗(2, k, km + w)-code exist, then a T ∗km+w (2, k, nk + w)-code also exists.

It is worth mentioning that a TD(k, n) is equivalent to a TD(k, n)−TD(k, m) with m = 0or 1. In what follows we will utilize this fact without mentioning it.

Now we proceed with our constructions of small T ∗(2, 6, v)-codes.

LEMMA 3.3. Both a T ∗1 (2, 6, 3)-code and a T ∗

3 (2, 6, 13)-code exist.

Proof. A T ∗1 (2, 6, 3)-code over alphabet Z3 can be obtained by taking the codewords:

(0,0,0,0,0,0), (1,1,1,1,0,2) and (2,2,2,2,0,1). A T ∗3 (2, 6, 13)-code is obtained by apply-

ing Lemma 3.1, making use of a 6-DGDD of type 26 from Lemma 2.2, and a T ∗1

(2, 6, 3)-code.

LEMMA 3.4. If v ∈ {18, 23, 33}, then a T ∗3 (2, 6, v)-code exists.

150 SHALABY ET AL.

Proof. A T ∗3 (2, 6, 18)-code follows from applying Lemma 3.1 with a 6-DGDD of type 36

which is constructed as follows:points: Z15 ∪ {x j : j = 1, 2, 3}groups: {x j : j = 1, 2, 3}, {i, i + 5, i + 10}, i = 0, 1, 2, 3, 4

blocks: (x1, 3, 0, 14, 7, 1) (x1, 8, 5, 4, 12, 6)

(x1, 13, 10, 9, 11, 2) (x2, 4, 1, 0, 8, 2)

(x2, 9, 6, 5, 13, 7) (x2, 14, 11, 10, 12, 3)

(x3, 5, 2, 1, 9, 3) (x3, 10, 7, 6, 14, 8)

(x3, 12, 0, 11, 13, 4) (6, 3, 2, 10, 4, x1)

(11, 8, 7, 9, 0, x1) (14, 2, 13, 6, 0, x2)

(13, 1, 12, 14, 5, x1) (7, 4, 3, 11, 5, x2)

(12, 9, 8, 10, 1, x2)

( j, j + 3, j + 6, j + 9, j + 12, x3) ( j = 0, 1, 2).

For v = 23, we take the alphabet as (GF(4)×{1, 2, . . . , 5})∪{A, B, C} and write (a, j)as a j where a ∈ GF(4) and 1 ≤ j ≤ 5. Then take the following codewords:

(14, 12, 01, (1 + x)5, x3, A) mod(4, −)

(A, 13, 15, 02, x4, x1) mod(4, −)

(14, (1 + x)1, x2, x5, 03, B) mod(4, −)

(B, 15, (1 + x)2, 04, x3, (1 + x)1) mod(4, −)

((1 + x)3, 05, 11, x4, (1 + x)2, C) mod(4, −)

(C, x1, (1 + x)4, (1 + x)3, 12, x5) mod(4, −)

((1 + x) j , x j , 1 j , 1 j , 1 j , 0 j ) ( j = 3, 4, 5)

(0 j , 0 j , 0 j , 1 j , x j , (1 + x) j ) ( j = 3, 4, 5)

(13, x2, 11, (1 + x)4, (1 + x)5, (1 + x)5)

(03, (1 + x)2, 01, x4, x5, x5)

((1 + x)3, (1 + x)3, 02, (1 + x)1, 14, 15)

(x3, x3, 12, x1, 04, 05)

((1 + x)5, (1 + x)1, (1 + x)2, (1 + x)3, (1 + x)4, (1 + x)4)

(x5, x1, x2, x3, x4, x4)

(02, 12, x2, x2, x2, (1 + x)2)

(05, 01, 02, 02, 03, 04)

((1 + x)1, (1 + x)1, x1, x1, 11, 01)

(01, 01, 11, 11, x1, (1 + x)1)

((1 + x)2, (1 + x)2, (1 + x)2, x2, 12, 02)

(15, 11, 12, 12, 13, 14)

where x is a primitive element of GF(4) satisfying x2 + x + 1 = 0, and the notationmod (4, −) after a codeword means that four codewords are obtained by adding to the firstcomponents all the elements of GF(4). In order to form a T ∗

3 (2, 6, 23)-code, we take threemore codewords over {A, B, C}, making use of a T ∗(2, 6, 3)-code from Lemma 3.3.

Forv = 33, we apply Construction 2.4, with a 6-DGDQD of type 5621 given in Lemma 2.3,to get the desired code.

PERFECT 4-DELETION-CORRECTING CODES 151

LEMMA 3.5. If v ∈ {63, 133, 153, 163, 183, 193}, then there exists a T ∗13(2, 6, v)-code.

Proof. For all stated values of v, except for 163, a B(6,1; (v − 1)/2) exists by Lemma1.6, and hence we have a 6-GDD of type 1

(v−13)

2 61. Inflate its points by 2 and replace everyblock by a 6-DGDD of type 26 which exists from Lemma 2.2. This yields a 6-DGDD oftype 2

(v−13)

2 121. Applying Lemma 3.1 with w = 1 gives the result, since a T ∗1 (2, 6, 3)-code

and T ∗(2, 6, 13)-code both exist by Lemma 3.3.For v = 163, note that a 6-GDD of type 513151 exists, which follows from a resolvable

B(5,1;65) (see, for example, [4]). Similar to the above construction, from this GDD we canobtain a 6-DGDD of type 1013301. We then apply Lemma 3.1 with w = 3, making use ofa T ∗

3 (2, 6, 13)-code and a T ∗3 (2, 6, 33)-code from Lemmas 3.3 and 3.4 respectively, to get

the result.

LEMMA 3.6. If v ∈ {93, 108}, then there exists a T ∗18(2, 6, v)-code.

Proof. Take a B(6,1;31) from Lemma 1.6 and regard it as a 6-GDD of type 12561, theninflate its points by 3. Since a 6-DGDD of type 36 exists from the proof of Lemma 3.4, wehave a 6-DGDD of type 325181. Start with a TD(6,9) and inflate its points by 2, making useof a 6-DGDD of type 26 from Lemma 2.2, we have a 6-DGDD of type 186. Thus, we canapply Lemma 3.1 with w = 0 to obtain a T ∗

18(2, 6, 93)-code and a T ∗18(2, 6, 108)-code. Here

the ingredients used are a T ∗(2, 6, 3)-code and a T ∗(2, 6, 18)-code, both of which exist byLemmas 3.3 and 3.4.

LEMMA 3.7. Ifv ∈ {8, 28, 38, 43, 53, 58, 83, 86, 87, 88, 99, 100, 101, 113, 115, 116, 117,

118, 143, 147, 148, 158}, then there exists a T ∗(2, 6, v)-code.

Proof. For all stated values of v, we take the alphabet to be Zv . The required codewordsare obtained by cycling modulo v the following base codewords.

v = 8: (0, 0, 5, 0, 1, 7)

v = 28: (1, 2, 5, 0, 10, 21) (16, 0, 6, 13, 0, 2)

v = 38: (5, 1, 15, 31, 18, 0) (0, 0, 2, 0, 6, 11)

(0, 27, 24, 18, 8, 1)

v = 43: (29, 7, 0, 5, 23, 8) (1, 0, 26, 35, 39, 7)

(26, 23, 0, 10, 12, 0)

v = 53: (10, 0, 37, 6, 8, 36) (42, 0, 1, 47, 10, 39)

(22, 39, 9, 0, 42, 4) (0, 0, 19, 0, 7, 32)

v = 58: (0, 35, 38, 25, 4, 10) (1, 6, 45, 42, 0, 2)

(18, 49, 41, 11, 2, 0) (5, 0, 12, 26, 0, 34)

v = 83: (52, 73, 76, 61, 0, 81) (43, 24, 0, 82, 77, 69)

(53, 27, 18, 0, 50, 11) (4, 66, 10, 70, 29, 0)

(47, 36, 48, 0, 2, 16) (0, 0, 28, 0, 43, 61)

152 SHALABY ET AL.

v = 86: (0, 55, 60, 4, 76, 44) (40, 38, 25, 77, 0, 74)

(66, 58, 22, 0, 3, 65) (39, 84, 64, 4, 57, 0)

(5, 0, 24, 38, 15, 32) (0, 0, 1, 58, 13, 69)

v = 87: (38, 0, 79, 11, 55, 80) (25, 2, 6, 16, 0, 59)

(10, 64, 71, 66, 0, 6) (14, 17, 0, 32, 45, 65)

(37, 0, 67, 75, 46, 72) (51, 1, 75, 0, 40, 0)

v = 88: (0, 28, 79, 86, 44, 76) (79, 17, 84, 0, 50, 30)

(69, 53, 0, 55, 84, 61) (6, 0, 75, 23, 43, 70)

(85, 22, 46, 15, 0, 60) (67, 0, 77, 1, 0, 41)

v = 99: (0, 27, 22, 8, 81, 82) (13, 0, 24, 20, 56, 96)

(65, 0, 15, 17, 5, 31) (71, 22, 69, 7, 0, 75)

(86, 80, 0, 45, 25, 48) (37, 29, 58, 1, 47, 0)

(0, 9, 66, 0, 78, 0)

v = 100: (6, 78, 80, 0, 21, 38) (97, 54, 67, 15, 22, 0)

(37, 48, 10, 0, 45, 79) (5, 0, 6, 89, 81, 93)

(0, 54, 10, 40, 36, 59) (1, 0, 51, 28, 67, 65)

(0, 29, 9, 53, 0, 0)

v = 101: (0, 39, 37, 13, 82, 14) (20, 0, 54, 72, 42, 51)

(97, 13, 61, 45, 0, 7) (42, 0, 5, 28, 20, 58)

(34, 0, 3, 100, 24, 60) (33, 39, 28, 0, 83, 74)

(17, 0, 0, 2, 10, 29)

v = 113: (0, 2, 65, 109, 94, 57) (0, 102, 51, 13, 101, 71)

(98, 59, 16, 33, 0, 23) (76, 97, 53, 79, 109, 0)

(9, 102, 68, 73, 7, 0) (88, 41, 17, 60, 14, 0)

(105, 0, 6, 41, 1, 28) (0, 0, 32, 0, 10, 78)

v = 115: (0, 30, 108, 93, 10, 104) (87, 0, 22, 31, 88, 8)

(10, 0, 61, 2, 26, 81) (12, 51, 9, 74, 111, 0)

(67, 24, 38, 0, 6, 82) (66, 40, 0, 69, 38, 21)

(0, 7, 40, 34, 53, 52) (0, 0, 5, 47, 90, 0)

v = 116: (22, 39, 58, 81, 0, 85) (0, 84, 8, 106, 86, 80)

(68, 12, 82, 23, 7, 0) (33, 19, 0, 87, 108, 16)

(0, 1, 66, 53, 92, 38) (66, 113, 0, 73, 64, 79)

(87, 0, 105, 20, 115, 32) (65, 0, 0, 5, 30, 74)

v = 117: (8, 63, 101, 68, 110, 0) (89, 64, 90, 109, 13, 0)

(0, 31, 90, 24, 56, 10) (0, 99, 94, 116, 11, 63)

(28, 0, 30, 106, 97, 100) (22, 12, 55, 99, 0, 113)

(113, 44, 94, 42, 0, 57) (38, 0, 35, 0, 6, 74)

PERFECT 4-DELETION-CORRECTING CODES 153

v = 118: (0, 13, 15, 107, 52, 69) (108, 0, 75, 24, 79, 47)

(100, 20, 8, 53, 0, 111) (113, 110, 41, 23, 0, 22)

(50, 37, 98, 0, 62, 40) (16, 89, 14, 35, 30, 0)

(82, 48, 42, 17, 0, 114) (0, 9, 44, 0, 50, 51)

v = 143: (0, 102, 91, 55, 122, 112) (90, 27, 67, 62, 53, 0)

(69, 132, 68, 35, 0, 86) (0, 130, 131, 105, 128, 47)

(105, 0, 83, 39, 66, 137) (24, 43, 113, 0, 58, 127)

(102, 114, 127, 8, 95, 0) (91, 136, 61, 0, 94, 43)

(0, 123, 73, 2, 8, 44) (0, 0, 4, 0, 9, 65)

v = 147: (0, 3, 37, 53, 115, 117) (0, 6, 55, 26, 67, 126)

(40, 55, 139, 50, 0, 73) (73, 71, 116, 54, 0, 13)

(25, 9, 14, 39, 79, 0) (10, 115, 100, 0, 139, 111)

(49, 26, 112, 105, 0, 109) (12, 31, 0, 48, 125, 69)

(125, 0, 88, 82, 7, 134) (52, 0, 1, 0, 28, 103)

v = 148: (0, 109, 106, 142, 95, 76) (122, 109, 131, 41, 137, 0)

(67, 55, 24, 69, 51, 0) (0, 128, 66, 32, 57, 126)

(106, 0, 99, 23, 41, 78) (85, 0, 116, 119, 21, 59)

(73, 97, 74, 144, 36, 0) (63, 46, 0, 19, 62, 92)

(0, 100, 8, 0, 13, 20) (50, 45, 94, 0, 35, 10)

v = 158: (0, 105, 145, 3, 71, 75) (0, 30, 125, 82, 140, 133)

(9, 31, 44, 0, 36, 122) (47, 147, 26, 12, 0, 106)

(132, 57, 146, 127, 0, 34) (51, 84, 2, 104, 0, 45)

(0, 18, 67, 7, 39, 17) (20, 62, 149, 68, 0, 64)

(61, 46, 108, 0, 29, 19) (0, 41, 104, 38, 66, 1)

(97, 0, 0, 24, 0, 81)

LEMMA 3.8. If v ∈ {68, 98, 128, 188}, then a T ∗(2, 6, v)-code exists.

Proof. We showed in Lemmas 3.3 and 3.4 that a T ∗3 (2, 6, u)-code exists for u ∈ {13, 18,

23, 33} and from Lemma 1.4 we know that a TD(6, u − 2) exists. Also from Lemma 3.7we have a T ∗(2, 6, 8)-code. Thus we apply Lemma 3.2 with n = u − 2, m = 1, w = 2 andk = 6 to get a T ∗(2, 6, 6(u − 3) + 8)-code.

LEMMA 3.9. If v ∈ {48, 73, 78, 103, 123, 138, 168, 198}, then a T ∗(2, 6, v)-code exists.

Proof. The conclusion comes from applying Lemma 3.2, where k = 6, m = 0 and w = 0or 1, with the following equations:

48 = 8 · 6, 73 = 12 · 6 + 1, 78 = 13 · 6,

103 = 17 · 6 + 1, 123 = 20 · 6 + 3, 138 = 23 · 6,

168 = 28 · 6, 198 = 33 · 6.

154 SHALABY ET AL.

The required TD(6, n)s for n ∈ {8, 12, 13, 17, 20, 23, 28, 33} follow from Lemma 1.4.The codes required as ingredients are all available from the previous lemmas, where aT ∗

1 (2, 6, n + 1)-code for n ∈ {12, 17} is obtained by replacing the subcode of a T ∗3 (2, 6,

n + 1)-code by a T ∗1 (2, 6, 3)-code.

LEMMA 3.10. If v ∈ {20, 21, 146}, then a T ∗(2, 6, v)-code exists.

Proof. For v ∈ {20, 21}, a T ∗(2, 6, v)-code based on the alphabet Zv can be obtained bytaking the following codewords.

v = 20: (2, 11, 0, 3, 10, 14), (12, 7, 4, 0, 11, 17),

(7, 1, 9, 2, 5, 15), (2, 1, 12, 13, 16, 18),

(6, 1, 1, 1, 3, 8), (6, 13, 11, 12, 12, 12),

(4, 13, 2, 2, 2, 2), (3, 3, 3, 3, 3, 19), (+4 mod 20).

v = 21: (1, 3, 5, 7, 0, 19), (11, 7, 1, 10, 20, 18),

(9, 12, 1, 2, 17, 8), (17, 14, 1, 1, 1, 1),

(2, 2, 2, 5, 6, 15), (3, 3, 3, 18, 4, 9), (+3 mod 21).

Now for v = 146, we first construct a T ∗2 (2, 6, 6)-code over {0, 1, 2, 3, a, b} by taking

the following codewords:

(0, 1, a, 1, 1, 2), (2, 3, a, 3, 3, 0), (3, 2, b, 2, 2, 1),(1, 0, b, 0, 0, 3), (b, b, a, a, b, b).

We then apply Lemma 3.1 with a 6-DGGD of type 46 given in Lemma 2.2 to get aT ∗

2 (2, 6, 26)-code. Finally, applying Lemma 3.2 with n = 24, m = 0, w = 2 and k = 6yields the desired code. Here the required TD(6, 24) comes from Lemma 1.4.

Summarizing the results of Lemmas 3.3–3.10, we have the following two theorems.

THEOREM 3.11. If v ∈ {20, 21, 86, 87, 99, 100, 101, 115, 116, 117, 146, 147}, then aT ∗(2, 6, v)-code exists.

THEOREM 3.12. If v ≡ 3 (mod 5), 3 ≤ v ≤ 208 and v �∈ {173, 178, 203, 208}, then aT ∗(2, 6, v)-code exists.

4. T∗(2, 6, v)-Codes with Large Values of v ≡ 3 (mod 5)

In this section, we establish the existence result of T ∗(2, 6, v) codes with v ≡ 3 (mod 5)and v ≥ 213 by applying the main construction in Section 2. It will be convenient for us toproceed in stages.

LEMMA 4.1. If v ≡ 3 (mod 5), 213 ≤ v ≤ 558, and v �∈ {318, 323, 328, 333, 458, 463,

468, 473, 478}, then a T ∗(2, 6, v)-code exists.

PERFECT 4-DELETION-CORRECTING CODES 155

Table 1.

30n + 5s + w + 2 n s w

213–243 7 0–6 1

248–278 8 0–6 6

273–313 9 0–8 1

338–388 11 0–10 6

363–418 12 0–11 1

393–453 13 0–12 1

483–558 16 0–15 1

Table 2.

6n + w 6m + w (n, m) w

318 18 (51, 1) 12

323 23 (52, 2) 11

328 28 (53, 3) 10

333 33 (54, 4) 9

458 8 (76, 1) 2

463 13 (77, 2) 1

468 18 (75, 0) 18

473 23 (76, 1) 17

478 28 (77, 2) 16

Proof. It is shown in Lemma 2.1 that for any n ∈ {7, 9, 12, 13, 16}, a DB(6, 1; 5n + 1)exists, and hence an IDB(6, 1; 5n + 1, 1) exists. Adding one extra point to 6-DGDDs oftype 59 and 512 in Lemma 2.2, we obtain an IDB(6, 1; 46, 6) and an IDB(6, 1; 61, 6).The result then follows from applying Lemma 2.5 with the parameters shown in Table 1.Here the required TDs and T ∗(2, 6, 5s + w + 2)-codes all exist by Lemma 1.4 andTheorem 3.12.

LEMMA 4.2. If v ∈ {318, 323, 328, 333, 458, 463, 468, 473, 478}, then a T ∗(2, 6, v)-codeexists.

Proof. Note that the T ∗(2, 6, 78)-code given in Lemma 3.9 contains a T ∗(2, 6, 13)-code asa subcode. Replacing this subcode with a T ∗

3 (2, 6, 13)-code we obtain a T ∗3 (2, 6, 78)-code.

From Lemmas 3.5 and 3.6, we know that both a T ∗13(2, 6, 63)-code and a T ∗

18(2, 6, 93)-code exist. Therefore, we establish the conclusion by applying Lemma 3.2 with parametersshown in Table 2. The required ITDs all exist by Lemmas 1.4 and 1.5.

Now we are able to give the main result of this section.

THEOREM 4.3. For all integers v ≡ 3 (mod 5) and v ≥ 213, a T ∗(2, 6, v)-code exists.

156 SHALABY ET AL.

Proof. For v ≤ 558 the conclusion holds by Lemmas 4.1 and 4.2. For v ≥ 563, it can bewritten as v = 30n + 5s + w + 2 where n, s and w are chosen so that:

(1) n ≡ 1, 3, or 5 (mod 6) and n ≥ 17, and hence a TD(7, n) exists;

(2) w = 1 if n ≡ 1 or 3 (mod 6), w = 6 if n ≡ 5 (mod 6);

(3) 5s + w + 2 ∈ {3, 8, 13, . . . , 63}.Lemmas 1.6 and 2.1 guarantee that an IDB(6, 1, 5n +w, w) exists for all values of n and

w chosen above. We can then apply Lemma 2.5 to obtain the result for v ≥ 563.

5. Conclusion

Combining the results of Theorems 3.11, 3.12 and 4.3 with the result of Theorem 1.1 givesTheorem 1.3, which is restated below.

THEOREM 5.1. For all positive integers v, there exists a T ∗(2, 6, v)-code, except possiblywhen v ∈ {173, 178, 203, 208}.

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