Geometric augmented product codes G. Altay, O.N. Ucan and H. Fatih U & gurda & g Abstract: We propose a new simple decomposable code construction technique that generates codes with the full information rate for all of the minimum Hamming distance-4 binary linear block codes of even length greater than or equal to eight. Additionally, some optimal Hamming distance-8 and higher distance codes are obtained with our proposed scheme. A generic trellis structure for the proposed codes was also designed. It is shown that our trellis structures provide lower decoding complexity in comparison to the trellises of some other well-known block codes. 1 Introduction Maximum-likelihood (ML) soft decision decoding of block codes using trellis decoding algorithms, such as BCJR [1] or Viterbi [2] , requires proper trellis structures. However, the main impediment in using trellis-based decoders for block codes is their complexity. For block codes, computationally efficient soft decision ML decoding can be achieved through low complex trellis structures and it is still a challenging problem. There are many studies in the literature that present different aspects of trellis complexity [3–6] . In order to achieve reduced trellis decoding complexity, decomposable block codes are a good alternative [7] .A code is said to be decomposable if it can be constructed by combining simple or shorter length component codes. If a code is decomposable, its trellis can also be decomposed into subtrellises [8]. Product codes, the ja þ xjb þ xja þ b þ xj Turyn construction [7] , squaring construction [9] , and generalised array codes (GAC) [10], are some of the well-known examples of decomposable code constructions. In this paper, to reduce decoding complexity and achieve bandwidth efficiency, we introduce a new construction scheme for decomposable codes that generates a wide range of linear block codes. This scheme was initially inspired by the ja þ xjb þ xja þ b þ xj Turyn construction. We name our proposed scheme as geometric augmented product (GAP) code construction. Basically, this construction employs a general single parity check matrix and two or more component generator matrices to geometrically construct a final product generator matrix for the constructed code. After specifying the generator matrices of component codes, GAP code construction sets geometric rules on how to place the component generator matrices in the final generator matrix to obtain a GAP code. By specifying suitable generator matrices for the component codes, GAP code construction guarantees to produce all of the Hamming distance-4 optimal binary linear block codes of even length greater than or equal to eight. Optimal code, in this context, means the code has the highest possible dimension (i.e. message length) among the codes of a given length and minimum distance. In other words, optimal code means a code with the full information rate. The optimal sizes for block codes can be looked up in the table of best-known codes [11] . This property makes GAP codes superior over GAC codes and the codes constructed in [12], since these methods can only construct some but not the entire full information rate distance-4 codes. GAP code construction can also produce some Hamming distance-8 optimal codes and some other high information rate codes of higher distances. Recently, extended Hamming codes have been shown to achieve near-Shannon-limit performance for very high rate coding, and that is useful for bandwidth efficient transmission [13]. However, extended Hamming codes have lengths of powers of 2, whereas our GAP codes have lengths of multiples of 2 and therefore provide great flexibility in adjusting the code length for a variety of applications. In order to decode GAP codes with low complexity, we have designed a well-structured generic trellis that consists of a set of structurally identical minimal trellises in parallel. These two features make our trellis efficient for decoder implementations. We show that Viterbi decoding of GAP codes requires smaller number of operations compared to some other well-known block codes. The main contributions of the paper are summarised as follows.  A novel generic generator matrix is proposed to obtain binary linear block codes with better code rates.  A new family of optimal Hamming distance-4 codes is derived from the proposed GAP code generator matrix. The length of codes in this family is even and greater than or equal to eight. We provide all the associated proofs for the size of this code family.  Some optimal and higher rate codes with Hamming distance greater than four are obtained using the proposed GAP code construction.  A generic trellis structure for GAP codes is designed and it is shown that our trellis structure provides lower decoding complexity compared to the trellises of some other well-known block codes when the Viterbi algorithm is used. E-mail: galtay@bahcesehir.edu.tr G. Altay and H. Fatih U& gurda& g are with Department of Electrical & Electronics Engineering, Bah- ce - sehir University, Bah- ce - sehir, Istanbul 34538, Turkey O.N. Ucan is with the Department of Electrical & Electronics Engineering, Istanbul University, Avcilar, Istanbul 34320, Turkey r The Institution of Engineering and Technology 2006 IEE Proceedings online no. 20050595 doi:10.1049/ip-com:20050595 Paper first received 1st November 2005 and in revised form 19th January 2006 IEE Proc.-Commun., Vol. 153, No. 5, October 2006 591