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The prior art construction of sets of balanced codewords by Knuth is attractive for its simplicity and absence of look-up tables, but the redundancy of the balanced codes generated by Knuth's algorithm falls a factor of two short with respect to capacity. We present a new construction, which is simple, does not use look-up tables, and is less redundant than Knuth's construction. In the new construction, the user word is modified in the same way as in Knuth's construction, that is by inverting a segment of user symbols. The prefix that indicates which segment has been inverted, however, is encoded and decoded in a different, more efficient, way.

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Let n be an arbitrary integer, let p be a prime factor of n. Denote by ! 1 the p t h primitive unity root, omega(1) : = e 2 pi i/p Define omega(i) : = omega 1(i) for 0 <= i <= p - 1 and B : = {1; omega 1, ... , omega(p-1)}(n) subset of C(n). Denote by K (n; p) the minimum k for which there exist vectors upsilon(1,) ... , upsilon(k) is an element of B such that for any vector omega is an element of B, there is an i, 1 <= i <= k, such that v(i) . omega = 0, where upsilon center dot omega is the usual scalar product of upsilon and omega. Grobner basis methods and linear algebra proof gives the lower bound K ( n; p) = n (p-1). Galvin posed the following problem: Let m = m ( n) denote the minimal integer such that there exists subsets Lambda(1,) ..., Lambda(m) of {1, ... , 4n} with vertical bar Lambda i vertical bar = 2n for each 1 <= i <= n, such that for any subset B subset of [4n] with 2 n elements there is at least one i, 1 <= i <= m, with A(i) boolean AND B having n elements. We obtain here the result m (p) >= p in the case of p > 3 primes.

Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature. I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to
avoid introducing new errors in the Second Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink
Rotterdam, November 2004

In 1986, Don Knuth published a very simple algorithm for constructing sets of bipolar codewords with equal numbers of 1s and 0s, called balanced codes. Knuth's algorithm is, since look-up tables are absent, well suited for use with large codewords. The redundancy of Knuths balanced codes is a factor of two larger than that of a code comprising the full set of balanced codewords. In our paper we will present results of our attempts to improve the performance of Knuths balanced codes.

In digital transmission systems, the transmission channel often does not pass d-c. This causes the well- known problem of baseline wander. One way to overcome this difficulty is to restrict the d-c content in the signal stream using suitably devised codes. It is shown that, for a d-c constrained code, the limiting efficiency is related to the number of allowable running digital sum states in a very simple way.

We derive the limiting efficiencies of dc-constrained codes. Given bounds on the running digital sum (RDS), the best possible coding efficiency η, for a K-ary transmission alphabet, is η = log2 λmax/log2 K, where λmax is the largest eigenvalue of a matrix which represents the transitions of the allowable states of RDS. Numerical results are presented for the three special cases of binary, ternary and quaternary alphabets.

Coding schemes in which each codeword contains equally many zeros and ones are constructed in such a way that they can be efficiently encoded and decoded.

A balanced code with r check bits and k information bits is a
binary code of length k+r and cardinality 2<sup>k</sup> such that each
codeword is balanced; that is, it has [(k+r)/2] 1's and [(k+r)/2] 0's.
This paper contains new methods to construct efficient balanced codes.
To design a balanced code, an information word with a low number of 1's
or 0's is compressed and then balanced using the saved space. On the
other hand, an information word having almost the same number of 1's and
0's is encoded using the single maps defined by Knuth's (1986)
complementation method. Three different constructions are presented.
Balanced codes with r check bits and k information bits with
k⩽2<sup>r+1</sup>-2, k⩽3×2<sup>r</sup>-8, and
k⩽5×2<sup>r</sup>-10r+c(r), c(r)∈{-15, -10, -5, 0, +5},
are given, improving the constructions found in the literature. In some
cases, the first two constructions have a parallel coding scheme

In a balanced code each codeword contains equally many 1's and
0's. Parallel decoding balanced codes with 2<sup>r</sup> (or 2<sup>r
</sup>-1) information bits are presented, where r is the number
of check bits. The 2<sup>2</sup>-r-1 construction given by D.E. Knuth
(ibid., vol.32, no.1, p.51-3, 1986) is improved. The new codes are shown
to be optimal when Knuth's complementation method is used

For n >0, d ⩾0, n ≡ d
(mod 2), let K ( n , d ) denote the minimal
cardinality of a family V of ±1 vectors of dimension
n , such that for any ±1 vector w of dimension
n there is a v ∈ V such that | v -
w |⩽ d , where v - w is the usual
scalar product of v and w . A generalization of a
simple construction due to D.E. Knuth (1986) shows that K ( n
, d )⩽[ n /( d +1)]. A linear algebra
proof is given here that this construction is optimal, so that
K ( n , d )-[ n /( d +1)] for all
n ≡ d (mod 2). This construction and its
extensions have applications to communication theory, especially to the
construction of signal sets for optical data links

Upper Bound on the Effi ciency of De-constrained Codes', Bell Sys t Positive Integer Powers of the Tri-diagonal Toeplitz Matrices

- T M Chien

T.M. Chien, ' Upper Bound on the Effi ciency of De-constrained Codes', Bell Sys t. Tech. J., vol. 49, pp. 2267-2287, Nov. 1970. [81 D.K. Salkuyeh, ' Positive Integer Powers of the Tri-diagonal Toeplitz Matrices', International Mathematical Forum, 1, no. 22, pp. 1061 -1065,

Design of some New Balanced CodesKnuth's Balancing of Codewords Revisited

- L G Tallini
- R M Capocelli
- B Bose

L.G. Tallini, R.M. Capocelli, and B. Bose, ' Design of some New Balanced Codes', IEEE Trans. Inform. Theory, vol. IT-42, pp. 790-802, May 1996. [61 HI. Weber and K.A S. Immink, 'Knuth's Balancing of Codewords Revisited', IEEE 1SIT2008 Conference, Toronto, July 2008.