Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2013, Article ID 625912, 5 pages http://dx.doi.org/10.1155/2013/625912

Research Article On Maximum Lee Distance Codes

Tim L. Alderson and Svenja Huntemann

Department of Mathematical Sciences, University of New Brunswick Saint John, Saint John, NB, Canada E2L 4L5

Correspondence should be addressed to Tim L. Alderson; [email protected]

Received 25 September 2012; Accepted 21 November 2012

Academic Editor: Annalisa de Bonis

Copyright © 2013 T. L. Alderson and S. Huntemann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Singleton-type upper bounds on the minimum Lee distance of general (not necessarily linear) Lee codes over Z𝑞 are discussed. Two boundsknownforlinearcodesareshowntoalsoholdinthegeneral case, and several new bounds are established. Codes meeting these bounds are investigated and in some cases characterised.

1. Introduction (Hamming) distance between any two codewords is 𝑑 (hence, no two codewords have as many as 𝑛−𝑑+1common The Lee metric was introduced by Lee1 [ ]in1958asan coordinates). Here, 𝑘=log𝑞|𝐶| is the dimension of 𝐶,which alternative to the Hamming metric for certain noisy channels. need not be an integer. Where context demands, we may also It found application and in particular was later developed denote the Hamming distance by 𝑑𝐻.TheSingletonbound for certain noisy channels (primarily those using phase-shift states that keying modulation [2]). The past decade has witnessed a burst of new and varied applications for codes defined in the 𝑑≤𝑛−⌈𝑘⌉ +1 (1) Lee metric (Lee codes) including constrained and partial- response channels [3], interleaving schemes [4], orthogo- and holds for all block codes. Codes meeting this bound nal frequency-division multiplexing [5], multidimensional with equality are called maximum distance separable (MDS) burst-error correction [6], and error correction for flash codes. Research on both linear and nonlinear MDS codes has memories [7]. These recent applications give increased inter- been extensive (e.g., see [8–10] and references therein). est in questions surrounding optimal Lee codes. Similar to the case of the Hamming metric, it is desirable 2.1. Lee Codes. Let Z𝑞 = {0,1,...,𝑞−1}be the set of repre- to investigate upper bounds on the minimum Lee distance sentatives of the integer equivalence classes modulo 𝑞.TheLee of a code given the code size, code length, and alphabet weight of any element 𝑎∈Z𝑞 is given by 𝑤𝐿(𝑎) = min{𝑎, 𝑞 − 𝑛 size. Codes meeting these bounds are of special interest as 𝑎}. Given an element 𝑐=(𝑐1,𝑐2,...,𝑐𝑛)∈Z𝑞,theLee weight they are optimal in the sense that their minimum distance is of 𝑐, denoted 𝑤𝐿(𝑐),isgivenby largest. Under the Hamming metric, such codes are referred to as maximum distance separable (MDS) codes. Under the 𝑛 𝑤 (𝑐) = ∑ {𝑎, 𝑞 − 𝑎} . Lee metric, such codes may be referred to as Maximum 𝐿 min (2) Lee Distance Separable (MLDS) codes. Here, we will present 𝑖=1 several upper bounds similar to the Singleton bound and 𝑛 For 𝑐, 𝑒 ∈ Z𝑞,theLee distance 𝑑𝐿(𝑐, 𝑒) between 𝑐 and 𝑒 is investigate the existence question of MLDS codes. In certain defined to be the Lee weight of their difference, cases, we are able to completely characterize MLDS codes. 𝑑𝐿 (𝑐, 𝑒) =𝑤𝐿 (𝑐−𝑒) 2. Preliminaries 𝑛 𝑘 (3) An (𝑛,𝑘,𝑑)𝑞 block code 𝐶 is a collection of 𝑞 𝑛-tuples (code- = ∑ min {𝑐𝑖 −𝑒𝑖 (mod 𝑞) ,𝑖 𝑒 −𝑐𝑖 (mod 𝑞)} . words) over an alphabet of size 𝑞 such that the minimum 𝑖=1 2 Journal of Discrete Mathematics

ALeecode𝐶 will be specified by (𝑛,𝑘,𝑑𝐿)𝑞,where𝑛,, 𝑘 Proposition 3. For any (𝑛,𝑘,𝑑𝐿)𝑞 Lee code 𝐶,thefollowing and 𝑞 are as aforementioned, and 𝑑𝐿 is the minimum Lee holds: distance of 𝐶;thatis,𝑑𝐿 = min{𝑑𝐿(𝑐, 𝑒) | 𝑐, 𝑒 ∈𝐶,𝑐 =𝑒}̸ . 𝑞=2 3 𝑞 Observe that in the case that or (binary or ternary 𝑑𝐿 ≤⌊ ⌋ (𝑛−⌈𝑘⌉ +1) . (7) codes), the Hamming and Lee metrics are identical. Thus a 2 code 𝐶 with 𝑞=2or 𝑞=3is MLDS if and only if 𝐶 is MDS. As much literature discusses MDS codes, we shall focus attention Remark 4. Observe that on the cases 𝑞>3. 𝑞 𝑑𝐿 −1 Acode𝐶 over the alphabet Z𝑞 is considered linear if 𝐶 𝑑 ≤⌊ ⌋ (𝑛−⌈𝑘⌉ +1) 󳨐⇒ ⌊ ⌋ 𝑛 𝐿 2 ⌊𝑞/2⌋ forms a submodule of Z𝑞.Unlessotherwisestated,wedonot assume linearity. ⌊𝑞/2⌋ − 1 (8) ≤⌊𝑛−⌈𝑘⌉ + ⌋ ⌊𝑞/2⌋ 2.2. Code Equivalence. Weshallsayapermutation 𝜎 of Z𝑞 is a 𝑑 (𝑎, 𝑏) = Lee permutation if it preserves Lee distance. That is, 𝐿 =𝑛−⌈𝑘⌉ . 𝑑𝐿(𝜎(𝑎), 𝜎(𝑏)),forall𝑎, 𝑏 ∈ Z𝑞. For example, any translation 𝜎(𝑥) = 𝑥 +𝑡(mod 𝑞) is a Lee permutation. Consequently, Proposition 3 subsumes the bound of Shiro- Given an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code 𝐶, we may define the following operations on codewords. A positional permutation moto (6). is a permutation (on 𝑛 letters) of the coordinate positions, Remark 5. Suppose that 𝐶 is an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code with applied to each word of 𝐶.ALee symbol permutation is ⌈𝑘⌉ = 1 meeting bound (7), so that 𝑑𝐿 =⌊𝑞/2⌋𝑛. Assume a Lee permutation applied to a fixed coordinate position that 𝐶 contains the zero codeword. It follows that 𝐶 contains throughout 𝐶. Two codes are equivalent if one may be precisely two codewords. Moreover, the nonzero codeword obtained from the other by applying a sequence of Lee symbol has (Lee) weight 𝑑𝐿.Hence,if𝑞 is even, then the nonzero or positional permutations. codeword has all entries 𝑞/2.If𝑞 is odd, then the nonzero (𝑞 + 1)/2 (𝑞 − 1)/2 Example 1. The code codeword has each entry being either or . We shall make improvements to bound (7) under various 𝐶=(0, 0)(1, 2)(2, 4)(3, 1)(4, 3) (4) settings. Observe that codes for which 𝑛<𝑘necessarily 𝑑 =0 (2,1,3) contain repeated codewords; whence, 𝐿 . Also, codes is a 5 Lee code. The equivalent codes may be produced 𝑛=𝑘 Z𝑛 (0)(14)(23) with necessarily comprise the entire space 𝑞,and by applying the Lee permutation to one or more 𝑑 =1 of the coordinate entries of 𝐶. For example, the code consequently 𝐿 .Hence,inthesequel,weshallfocuson codes with 𝑛>𝑘. 󸀠 𝐶 = (0, 0)(1, 3)(2, 1)(3, 4)(4, 2) (5) 3.1. Bounds on Codes over Even-Sized Alphabets. By restrict- is equivalent to 𝐶. ing to codes over even-sized alphabets, we obtain the follow- ing bound. It follows that any given Lee Code is equivalent to a Lee Code containing the zero codeword. This observation shall Theorem 6. For any Lee metric code with alphabet size 𝑞 even, prove quite useful in the sequel. of length 𝑛 and minimum Lee distance 𝑑𝐿,thefollowingholds: 𝑞 𝑑 ≤ 𝑛−⌈ |𝐶|⌉+1. 3. Singleton-Type Bounds 𝐿 2 log2 (9) In 2000 Shiromoto, [11]provedthefollowingupperboundfor Proof. If 𝑞 is even, we can create a distance preserving map linear codes over Z𝑞. (Gray code) 𝜙 (similar to that used in [12, 13]) from the metric (Z ,𝑑 ) (Z𝑞/2,𝑑 ) Proposition 2 (see [11]). If 𝐶 is a linear (𝑛,𝑘,𝑑𝐿)𝑞 Lee code space 𝑞 𝐿 to the metric space 2 𝐻 defined by over Z𝑞,then 𝑞/2−𝑎 𝑎 𝑞 {⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞ 𝑑 −1 { 000 ⋅ ⋅ ⋅ 111 if 0≤𝑎≤ −1, 𝐿 𝜙 (𝑎) = 2 ⌊ ⌋≤𝑛−𝑘. (6) { 𝑞−𝑎 𝑎−𝑞/2 (10) ⌊𝑞/2⌋ {⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞ 𝑞 111 ⋅ ⋅ ⋅ 000 ≤𝑎≤𝑞−1. { if 2 We shall show that bound (6) holds for general (not For 𝑐∈𝐶, we extend the map coordinatewise; thus, 𝜙(𝑐) = necessarily linear) Lee codes. Indeed, observe that for a given 󸀠 𝑑 (𝜙(𝑐1)⋅⋅⋅𝜙(𝑐𝑛)), and let 𝐶 = 𝜙(𝐶) = {𝜙(𝑐) |𝑐∈𝐶}.Itfollows code, any two codewords differ in at most 𝐻 coordinate 󸀠 󸀠 𝑑 ≤⌊𝑞/2⌋𝑑 that 𝐶 is a binary code of length 𝑛(𝑞/2) with |𝐶| = |𝐶 |,and positions, from which it follows that 𝐿 𝐻. 󸀠 Combining this observation with the Singleton bound (1) 𝑑𝐿(𝐶) =𝐻 𝑑 (𝐶 ).TheresultfollowsfromtheSingletonbound gives the following. (1). Journal of Discrete Mathematics 3

3.1.1. Special Cases: Characterizing and Improving. If restric- 3.2. Bounds on General Alphabets. In this section, we shall tions are made on the size of 𝐶, we are able to characterize make improvements to the bound in Proposition 3.First, cases of equality in bound (9). The following well known some intermediary lemmata are introduced. property of binary MDS codes shall be of use. See, for example, [14]. Lemma 11. If 𝐶 is an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code with 𝑑𝐿 ≥ ⌊𝑞/2⌋(𝑛 − ⌊𝑘⌋) +, 1 then the following hold: Lemma 7. If 𝐶 is a binary (𝑛,𝑘,𝑑)2 MDS code of integral 𝑘 dimension 𝑘≥1, then either (1) is an integer; (2) Fix 𝑘 coordinate positions 𝑖1 <𝑖2 <⋅⋅⋅<𝑖𝑘,and𝑘 (not (1) 𝑘=1,and𝐶 is the repetition code {(0,0,...,0), necessarily distinct) elements 𝛼1,...,𝛼𝑘 ∈ Z𝑞.Then, (1,1,...,1)}, thereexistspreciselyonecodeword𝑐 having 𝑐𝑖 =𝛼𝑗, 𝑘≥2 𝑛=𝑘+1 𝐶 𝑗 (2) , ,and is the binary parity check code 𝑗=1,2,...,𝑘. consisting of all binary 𝑛-tuples of even (Hamming) weight, or Proof. For the first part, observe that if two codewords 𝑛 ⌊𝑘⌋ (3) 𝑘≥2, 𝑛=𝑘,and𝐶 is the entire space Z2. agree in coordinates, then their distance is at most ⌊𝑞/2⌋(𝑛−⌊𝑘⌋),violatingtheminimumLeedistancecondition. ⌊𝑘⌋ Lemma 8. Let 𝐶 be an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code with 𝑛>𝑘≥1, Consequently, |𝐶| ≤ 𝑞 ,giving𝑘=⌊𝑘⌋. 𝑞>2 |𝐶| ∈ Z 𝑑 =(𝑞/2)𝑛− |𝐶| + 1 even, and log2 .If 𝐿 log2 ,then For the second part, fix 𝑘 coordinate positions {𝑖1,...,𝑖𝑘}. either The result then follows by observing that no two codewords 𝑞𝑘 (1) 𝑘=1,and𝐶 is the repetition code agreeinallofthesepositions,andthereare codewords in {(00⋅⋅⋅0) ((𝑞/2)(𝑞/2)⋅⋅⋅𝑞/2)},or total. 𝑘 = 𝑛 − (1/2) 𝑞=4 (2) ,and . Lemma 12. If 𝐶 is a (𝑘+1,𝑘,3)5 Lee code, 𝑘∈Z,then𝑘=1. 𝑑 =(𝑞/2)𝑛− |𝐶| + 1 𝜙 Proof. Assume that 𝐿 log2 .Let be as Proof. First, note that from Example 1,a(2, 1, 3)5 Lee code 󸀠 defined in the proof of Theorem 6.Then,𝐶 = 𝜙(𝐶) is a binary does in fact exist. 󸀠 󸀠 󸀠 󸀠 󸀠 (𝑛 ,𝑘 ,𝑑 ) MDS code, 𝑛 =(𝑞/2)𝑛,and𝑘 = log2|𝐶|.Notethat Now, let 𝑘≥2, and assume without loss of generality that 󸀠 since 𝑛≥2and 𝑞≥4,wehave𝑛 ≥4. According to Lemma 7, thezerocodewordisin𝐶. Consider (Lemma 11)thethree we have three cases to consider. codewords 𝑤1, 𝑤2,and𝑤3,havingfirst𝑘 coordinates 100⋅⋅⋅0, 0100 ⋅⋅⋅0,and400 ⋅ ⋅ ⋅ 0, respectively. In order to maintain 󸀠 󸀠 Case 1 (𝑘 =1).Inthiscase,𝐶 is the binary repetition code; minimum Lee distance 3 from the zero codeword, the (𝑘+1)- hence, 𝐶 is the code st coordinate of each 𝑤𝑖 must be from {2, 3}.Thus,atleast 𝑞 𝑞 𝑞 two of {𝑤1,𝑤2,𝑤3} agree in the final coordinate, violating the (00⋅⋅⋅0) ( ⋅⋅⋅ ). 2 2 2 (11) minimum distance property.

Lemma 13. If 𝐶 is an (𝑛, 𝑘,𝐿 𝑑 )𝑞 code with 𝑞>3,andintegral 𝑘>1 𝑑 ≤ ⌊𝑞/2⌋(𝑛 − 𝑘) 󸀠 󸀠 󸀠 󸀠 dimension ,then 𝐿 . Case 2 (𝑘 >1,𝑛 =𝑘+1).Inthiscase,𝐶 is the binary parity 𝑛󸀠 (101000 ⋅⋅⋅0) check code of length .Assuch,thevector is Proof. Suppose that 𝐶 is an (𝑛,𝑘,𝑑𝐿)𝑞 code, 𝑞>3, 1<𝑘∈ 𝐶󸀠 𝜙 𝑞/2 ≤ 2 in . From the definition of ,itfollowsthat ;so, Z,with𝑑𝐿 ≥ ⌊𝑞/2⌋(𝑛 − 𝑘).From +1 Lemma 11, there exist 𝑞=4. 𝑎 𝑏, 𝑐 𝑘−1 0 𝑘 󸀠 󸀠 codewords , and having first coordinates and th Note that since 𝑞=4,wehave𝑛 =2𝑛=𝑘 +1=2𝑘+1. coordinate 0, 1,and𝑞−1,respectively.Consequently,wehave 𝑑𝐿 = ⌊𝑞/2⌋(𝑛 − 𝑘). +1 󸀠 󸀠 󸀠 󸀠 𝑘󸀠 Case 3 (𝑘 >1,𝑛 =𝑘).Inthiscase,𝐶 = Z2 .Again,the If 𝑞 is even, then 𝑑𝐿(𝑎, 𝑏) = (𝑞/2)(𝑛 −𝑘)+1 gives 𝑏𝑖 = 󸀠 󸀠 vector (101000 ⋅ ⋅ ⋅ 0) is in 𝐶 ,giving𝑞=4.Butthen,𝑛 =2𝑛= 𝑎𝑖 +(𝑞/2)(mod 𝑞),for𝑖 = 𝑘+1,𝑘+2,...,𝑛.Similarly,𝑐𝑖 =𝑎𝑖 + 󸀠 𝑘 =2𝑘, contradicting 𝑛>𝑘. (𝑞/2) (mod 𝑞),for𝑖=𝑘+1,𝑘+2,...,𝑛.Butthen,𝑑𝐿(𝑏, 𝑐), =2 contradicting the condition 𝑑𝐿 = (𝑞/2)(𝑛 − 𝑘). +1 Remark 9. An example of a code satisfying item (2) in If 𝑞 is odd, then a similar argument shows that for each Lemma 8 is the (2, 3/2, 2)4 Lee code: of the final (𝑛 − 𝑘) coordinate, 𝑑𝐿(𝑏𝑖,𝑐𝑖)≤1.Consequently, 𝑑𝐿(𝑏, 𝑐) ≤ 𝑛−𝑘+2. Therefore, we have 𝑛−𝑘+2 ≥ ⌊𝑞/2⌋(𝑛−𝑘)+1 𝐶= 00 , 02 , 20 , 22 , 13 , 31 , 11 , 33 . {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} (12) from which it follows that 𝑞=5, 𝑛=𝑘+1,and𝑑=3.From Lemma 12,itthenfollowsthat𝑘=1, a contradiction. With Lemma 8, we see that the bound in Theorem 6 may be improved in certain cases. From Lemmas 11 and 13,wehavethefollowingresult. Theorem 10. 𝐶 (𝑛,𝑘,𝑑 ) 𝑞>4 Let be an 𝐿 𝑞 Leecodewith even, Theorem 14. If 𝐶 is an (𝑛, 𝑘,𝐿 𝑑 )𝑞 Lee code 𝑞>3, 𝑛>𝑘>1 𝑛>𝑘≥1,andlog2|𝐶| ∈ Z.Then, then 𝑞 𝑞 𝑑 ≤ 𝑛− |𝐶| . 𝑑 ≤⌊ ⌋ (𝑛−⌊𝑘⌋) . 𝐿 2 log2 (13) 𝐿 2 (14) 4 Journal of Discrete Mathematics

3.3. Bound Based on Plotkin’s Average Distance Bound. Based Remark 18. Chiang and Wolf [16]establishedtheboundin onthefactthattheminimumdistancebetweenpairsof Corollary 17 forlinearcodes. codewords cannot exceed the average distance between all pairsofdistinctcodewords,WynerandGraham[15]obtained Codes meeting the bounds in Theorem 16 do exist [16]. the following bound. Two examples follow: (1) for 𝑝 as prime, the linear one-dimensional (equidis- Proposition 15. If 𝐶 is an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code, then tant) (𝑝 − 1, 1, ((𝑝 − 1)/2) ⋅3)𝑝 code generated by 𝑛𝐷 (1,2,...,𝑝−1); 𝑑 ≤ , (15) 𝐿 1−(1/ |𝐶|) (2) the linear (4, 2, 4)5 Lee code with generator matrix where 𝐷 is the average Lee weight of Z𝑞,givenby 3410 𝐺=[ ]. 2 0341 (22) {𝑞 −1 { if 𝑞 is odd, 𝐷= 4𝑞 {𝑞 (16) 4. Conclusion { 𝑞 . {4 if is even Several upper bounds on the minimum Lee distance of a Using the previous result, we may establish a further block code similar to the Singleton bound were established. Singleton-type bound for Lee codes. Codes actually meeting some of these bounds were pre- sented. Two bounds known for linear codes were shown (see Theorem 16. Let 𝐶 be an (𝑛, 𝑘,𝐿 𝑑 )𝑞 code. Let 𝛼 be the maximal ⌈𝑘⌉ − 1 Proposition 2 and Corollary 17) to hold for nonlinear codes. number of codewords mutually agreeing in common Several new bounds were also established. coordinates. Then, 𝑑 ≤ ⌊(𝑛−⌈𝑘⌉ +1) ⋅𝑎⌋ , 𝐿 (17) Acknowledgments where 2 The first author thanks the NSERC of Canada for support {𝑞 −1 𝛼 { ⋅ 𝑖𝑓 𝑞 odd, through the Discovery Grants Program. The second author 𝑎= 4𝑞 𝛼−1 {𝑞 𝛼 (18) thanks the NSERC of Canada for support through the { ⋅ 𝑖𝑓 𝑞 . Undergraduate Student Research Awards Program. {4 𝛼−1 even 𝐶 (𝑛,𝑘,𝑑 ) 𝛼 Proof. Let be an 𝐿 𝑞 code, and let be as defined References earlier. Consider a collection 𝐶1 of 𝛼 codewords mutually agreeing in ⌈𝑘⌉−1 coordinates, assumed to be the first ⌈𝑘⌉−1 [1] C. Y. Lee, “Some properties of nonbinary error-correcting 󸀠 coordinates. Let 𝐶 be the corresponding punctured code codes,” IEEE Transactions on Information Theory,vol.4,no.2, obtained by deleting the first ⌈𝑘⌉−1 coordinates of each word Article ID 012263, pp. 77–82, 1958. in 𝐶1. [2] K. Nakamura, “A class of error correcting codes for DPSK 󸀠 󸀠 󸀠 It follows that 𝐶 is an (𝑛−⌈𝑘⌉+1,1,𝑑 𝐿)𝑞 code with 𝑑 𝐿 ≥ channels,” in Proceedings of the International Conference on 󸀠 Communications (ICC ’79),vol.3,p.45,1979. 𝑑𝐿. Applying Proposition 15 to 𝐶 ,weobtain [3] R. M. Roth and P. H. Siegel, “Lee-metric BCH codes and their 󸀠 (𝑛−⌈𝑘⌉ +1) 𝐷 application to constrained and partial-response channels,” IEEE 𝑑 ≤𝑑 ≤ . (19) 𝐿 𝐿 1−(1/𝛼) Transactions on Information Theory,vol.40,no.4,pp.1083– 1096, 1994. Finally, applying (16), we get the result. [4] M. Blaum, J. Bruck, and A. Vardy, “Interleaving schemes for multidimensional cluster errors,” IEEE Transactions on Infor- Suppose that 𝐶 is an (𝑛,𝑘,𝑑𝐿)𝑞 Lee code, and 𝛼 is as ⌈𝑘⌉−1 mation Theory, vol. 44, no. 2, pp. 730–743, 1998. definedealier.Observethatsince|𝐶| > 𝑞 ,itfollowsthat 𝛼≥2 𝑞 [5] K.-U. Schmidt, “Complementary sets, generalized Reed-Muller .So,if is even, the bound in Theorem 16 always meets codes, and power control for OFDM,” IEEE Transactions on 𝑘∈Z or exceeds bound (7). Moreover, if , simple counting Information Theory, vol. 53, no. 2, pp. 808–814, 2007. 𝛼≥𝑞 shows that . This gives the following corollary, much [6] T.Etzion and E. Yaakobi, “Error-correction of multidimensional improving bound (7). bursts,” IEEE Transactions on Information Theory,vol.55,no.3, pp.961–976,2009. Corollary 17. 𝐶 (𝑛,𝑘,𝑑 ) 𝑘∈Z If is an 𝐿 𝑞 Lee code with ,then [7] A. Barg and A. Mazumdar, “Codes in permutations and error correction for rank modulation,” IEEE Transactions on Informa- 𝑑𝐿 ≤ (𝑛−𝑘+1) ⋅𝑎, (20) tion Theory,vol.56,no.7,pp.3158–3165,2010. where [8] J. W. P. Hirschfeld, G. Korchmaros, and F. Torres, Algebraic 2 Curves over a Finite Field, Princeton Series in Applied Mathe- { 𝑞 { 𝑞 , matics, Princeton University Press, Princeton, NJ, USA, 2008. {4(𝑞−1) if even 𝑎={ (21) [9] T. L. Alderson, A. A. Bruen, and R. Silverman, “Maximum {𝑞+1 if 𝑞 odd. distance separable codes and arcs in projective spaces,” Journal { 4 of Combinatorial Theory A,vol.114,no.6,pp.1101–1117,2007. Journal of Discrete Mathematics 5

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[12] S. T. Dougherty and C. Fernandez-Cordoba, “Codes over Z2𝜅 , Gray map and self-dual codes,” Advances in Mathematics of Communications, vol. 5, no. 4, pp. 571–588, 2011. [13] S. T. Dougherty and K. Shiromoto, “Maximum distance codes over rings of order 4,” IEEE Transactions on Information Theory, vol.47,no.1,pp.400–404,2001. [14]F.J.MacWilliamsandN.J.A.Sloane,The Theory of Error- Correcting Codes. I,vol.16ofNorth-Holland Mathematical Library, North-Holland Publishing, Amsterdam, The Nether- land, 1977. [15] A. D. Wyner and R. L. Graham, “An upper bound on minimum distance for a k-ary code,” Information and Control,vol.13,no. 1,pp.46–52,1968. [16] J. C. Chiang and J. K. Wolf, “On channels and codes for the Lee metric,” Information and Control,vol.19,no.2,pp.159–173,1971. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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