text: by way of introduction of a detailed description of preferred embodiments of the arrangement described herein invention , some of the theoretical principles underlying such an arrangement will now be briefly discussed by way of direct comparison with the related art described in the foregoing . as a first point , the min - sum ( ms ) approximation will be shown to be a straightforward simplification of the check - node computation . the reliability of the messages coming out of a check - node update can be expected to be dominated by the least reliable incoming message . the ms outputs are , in modulus , slightly larger than those output by a non - approximated check - node processor . this results in a significant error rate degradation . for this reason , chen et al . ( already cited in the foregoing ) have proposed to resort to normalized - ms ( n - ms ) to partially compensate for these losses : n - ms typically consists of a simple multiplication of the output messages by a scaling factor . the factor can be optimized through simulations or , in a more sophisticated way , with density evolution as disclosed by chen et al . this approach recovers most of the performance gap caused by ms and makes ms a valid alternative to a full processing approach . an almost equivalent alternative to the n - ms is the offset - min - sum ( o - ms ), again disclosed by chen et al ., that performs slightly worse than n - ms . a ms decoder does not require knowledge of the noise variance , which is of great interest when the noise variance in unknown or hard to be determined . more sophisticated approximations are able to perform nearly the same as a full precision approach , but generally require a data dependent correction term that makes the check - node processor more complex . this specific issue has been investigated in the art ( see , e . g ., zarkeshvari , f . banihashemi , a . h . : on implementation of min - sum algorithm for decoding low - density parity - check ( ldpc ) codes : globecom &# 39 ; 02 . ieee vol . 2 , 17 - 21 november 2002 , pp . 1349 - 1353 ). parallel or partially parallel architectures employ a multiplicity of check - node processors . for this reason any simplification of this computation kernel is of particular interest . when ms is adopted , the same modulus is shared by all outgoing messages from a check - node update processor ; its value is equal to the smaller modulus among the incoming messages . the only exception is the outgoing message that corresponds to bit whose incoming massage has the smaller modulus . the modulus of such outgoing message is equal to the second smaller among the incoming messages . hence , the minimum check - to - bit information to be stored is much less in comparison with the approaches described so far . for that reason , normalized ms approximation , with a memory efficient approach , is proposed here in conjunction with the layered decoding ( l - spa ) to compensate for the ms performance degradation thanks to the faster convergence given by the scheduling modification . while a more detailed analysis of the storage capability will be provided in the following , with a detailed comparison with the other cases , it will noted that , by adopting the approach described herein , storing ( i ) two moduli ; ( ii ) the signs of all the outgoing messages ; ( iii ) the position of the least reliable message will suffice . the new approach is capable of outperforming conventional spa with the same number of iterations , while requiring about 70 % less memory . the approach considered here ( which may be designated layered - normalized - min - sum , i . e ., l - n - ms ) applies a memory efficient normalized min - sum approach to a layered decoding schedule is schematically represented below . λ j = λ j ∀ j for k = 1 : n ite for i = 1 : nc for j ∈ v ( i ) if j ≠ m ( i ) q ~ ji = λ j - r i 1  s ij else q ~ ji = λ j - r i 2  s ij r i 1 = min   q ~ ji   /  α m  ( i ) = arg j   min   q ~ ji  r i 2 = min j ≠ m  ( i )   q ~ ji   /  α for j ∈ v ( i ) s ij = ( sign   ( q ~ ji )  •  ∏ m ∈ v  ( i )   sign   ( q ~ mi ) ) if j ≠ m ( i ) λ j = q ~ ji + r i 1  s ij else λ j = q ~ ji + r i 2  s ij where r i 1 and r i 2 , are the smallest and second smallest check - to - bit message modulus , m ( i ) is the least reliable bit in equation i , s mi are the signs of the outgoing messages and α is the scaling factor of n - ms . performance of the l - m - ms proposed herein can be compared with performance achievable with : a layered decoding and pure ms ( i . e ., without normalization factor ) ( l - ms ); with layered decoding algorithm ( l - spa ); and with a conventional spa . for instance a meaningful comparison can be performed at 25 iterations . as a first example , a structured ldpcc code , designed by the team of prof . wesel ( university of california los angeles ) has been used for the comparison . code is designed with same graph conditioning adopted in vila casado a . i . ; weng w . ; wesel r . d . : “ multiple rate low - density parity - check codes with constant block length ”, asilomar conf . on signals , systems and computers , pacific grove , calif ., 2004 . the code is 1944 bits long with rate ⅔ . it is designed with a combination of 8 × 24 = 192 cyclically shifted identity matrices and null matrices of size 81 × 81 . the number of edges is equal to 7613 with maximum variable degree equal to 8 and maximum check degree equal to 13 . the parity part is organized as described in fig4 . the upper right matrix d is defined ( parity section only ) by eq 15 below for a rate ⅔ code structure . the results show l - n - ms performs slightly better than conventional spa , but requires much simpler check - node processing and a dramatically smaller amount of memory . the gap between l - spa and l - ms is mostly recovered by means of the normalization factor . the normalization factor α has been optimized through simulations focusing on frame error rate — fer equal to 10 − 2 with the resulting value equal to 1 . 35 . as a second example , a high rate structured ldpcc code of similar size has been selected among those proposed in eleftheriou e . ; ölcer s . : low density parity - check codes for digital subscriber lines , in proc ., icc &# 39 ; 2002 , new york , n . y ., pp . 1752 - 1757 . the code has a linear encoding complexity and supports layered decoding . it is 2209 bits long and it has rate 0 . 9149 . in this case l - n - ms performs even slightly better than the l - spa . an explanation could be found in the code structure that may have more short cycles compared to the previous example , so that spa becomes less efficient . the normalization factor α was equal to 1 . 3 . fixed - point implementation of n - ms would require a multiplication by a factor with a high accuracy in the quantization level and a significant complexity due to the operator itself . however , it is possible to simplify the normalization procedure at the cost of negligible performance loss . the normalization can be implemented very efficiently with the following approach : where the operator ( x & gt ;& gt ; y ) represent a y bits right shift of message x . for both examples s has been chosen equal to 2 , that corresponds to a = 1 . 333 . one may define a uniform quantization scheme ( n b , p ), where n b is the number of bits ( including sign ) and p is the fraction of bits dedicated to the decimal part ( i . e ., the quantization interval is 2 − p ). the adopted quantization schemes are the best for a given number of bits n b . for the rate ⅔ code not even 8 bits are sufficient to perform close to the floating point precision . however , if the same quantization scheme is applied to decode a similar rate ⅔ code with size 648 bits , it results that l - n - ms with ( 8 - 4 ) performs better than floating point spa at 12 iterations . this result is consistent with the results reported in zarkeshvari et al . ( already cited ), where it has been noted that the ms approximation works pretty well with short codes and quantized messages . for the higher rate code even 6 bits were found to lead to negligible losses . the n - ms approach allows a significant reduction of the memory to store the check - to - bit messages r ij . in fact , the amount of memory turns out to be : ( i ) 2 * nc *( n b − 1 ) bits for the modulus of the two least reliable check - to - bit messages of each check ( where nc is the number of checks ); ( ii ) the sign of all check - to - bit messages that result in e bits ; ( iii ) the position of the least reliable message in the check that results in nc * ceil ( log2 ( dc )) bits , where dc is ( maximum ) check - node degree , and [ ceil ] denotes the ceiling operator . table 2 below summarizes the results of comparison of the memory requirements for the approaches presented so far . specifically , table 2 refers to the memory needed to store the messages r ij and q ij and reports the results of comparison between conventional check - node and memory efficient ms approximation applied to different decoding algorithms . e * n b + 2 * nc * ( n b − 1 ) + e + ceil ( log2 ( dc )) 2 * n * n b + 2 * nc * ( n b − 1 ) + e + ceil ( log2 ( dc )) n * n b + 2 * nc * ( n b − 1 ) + e + ceil ( log2 ( dc )) the results in terms of memory requirements for the simulated codes indicate that the l - n - ms approach proposed herein requires 70 % and 76 % less memory than the conventional implementations of the spa algorithm for rate ⅔ code and rate 0 . 9149 code , respectively . at the cost of some minor performance losses , memory requirements can be reduced by a factor 24 %, 42 % and 50 % when the memory efficient ms solution is applied to spa , m - spa , and l - spa , respectively , for the rate ⅔ code considered . for the rate 0 . 9149 code , the reduction amounts to 24 %, 51 % and 61 %. a “ memory efficient ” ms entails some significant , potential advantages that relate to the implementation of high - speed parallel decoders . a first advantage lies in that a check - node requires much less input / output bits , so that routing problems can be scaled - down compared to a conventional approach . secondly , in vectorized decoders explicitly dedicated to structured ldpcc ( see , novichkov et al . and wo - a - 02 / 103631 — both already cited ), memory paging is designed so that all messages belonging to the same non - null sub - block in the parity check matrix are stored in the same memory word . a switch - bar is then adopted to cyclically rotate the message after / before the r / w operation . the approach discussed herein provides for the possibility of implementing switch - bars for a only . fig5 is a functional block diagram of an embodiment of a decoder . with reference to the general layout of fig1 , the decoder 20 is intended to be located downstream of the demodulator 18 to produce decoded data 22 . the decoder 20 receives as its input the llr values produced by the demodulator 18 ( the demodulator may be implemented in a way to provide these values directly ). the decoder 20 processes these llr to retrieve the decoded data 22 . referring to fig5 , the decoder 20 is configured to receive from the demodulator 18 initial values ) λ j for initialization ( i . e ., λ j = λ j for each j ) and to produce as an output from a memory block designated a the messages λ j which are representative of the a - posteriori probability of the output decoded data . specifically , the decoder receives as its input the logarithm of the ratio of the likelihood for each bit , i . e ., λ j ; the decoder yields λ j , i . e ., the logarithm of the ratio of the a - posteriori probabilities . the decoder 20 herein is assumed ( just by way of example , with no intended limitation of the scope of the invention ) to operate with “ parallelism 3 ”, i . e ., a structured ldpcc with subblock size equal to 3 is assumed . the basic layout of the arrangement implemented in the decoder of fig5 is repeated below for immediate reference . λ j = λ j ∀ j for k = 1 : n ite for i = 1 : nc for j ∈ v ( i ) if j ≠ m ( i ) q ~ ji = λ j - r i 1  s ji else q ~ ji = λ j - r i 2  s ij r i 1 = min   q ~ ji  / α m  ( i ) = arg j   min   q ~ ji  r i 2 = min j ≠ m  ( i )   q ~ ji  / α for j ∈ v ( i ) s ij = ( sign  ( q ~ ji ) · ∏ m ∈ v  ( i )  sign  ( q ~ mi ) ) if j ≠ m ( i ) λ j = q ~ ji + r i 1  s ij else λ j = q ~ ji + r i 2  s ij where r i 1 and r i 2 are the smallest and second smallest check - to - bit message modulus , m ( i ) is the least reliable bit in equation i , s mi are the signs of the outgoing messages and α is the scaling factor of n - ms . the memory block designated a stores the messages λ j ; each word contains the values belonging to three consecutive bit nodes . the memory block designated s stores the signs s ij ; three signs belonging to three consecutive messages └ s 3i , 3j s 3i + 1 , 3j + 1 s 3i + 2 , 3j + 2 ┘ are arranged together to form a memory word . the memory block designated r contains three messages related to the minimum and second minimum and minimum position , i . e ., the memory block designated r contains three messages related to i ) the value of the minimum , ii ) the value of the second minimum and iii ) the minimum position . the messages are arranged together in such a way that all the messages related to the check equations that must be run in parallel ( a super - code ) can be read simultaneously ; an example of memory word content is given below : the input messages to the memory block a and the output messages therefrom are rotated back and forward according to the proper shift values . in the embodiment shown herein , this function is performed via switch - bars 100 , 102 arranged at the input and the output of the memory block a . the messages coming out of the memory blocks a , s , and r are demultiplexed towards the proper blocks q configured to perform the computation of the values { tilde over ( q )} ji in the embodiment shown herein , the demultiplexing is performed via three demultiplexers 104 , 106 , and 108 each serving a respective one of three blocks q . as illustrated , a bit - to - check module 120 comprises a plurality of bit - to - check generators q . the three blocks q in turn feed a corresponding block cnp ( check node processor ). the cnp blocks are configured to perform the following functions : i ) the search of the minimum , its position and the second minimum ( r i 1 ; r i 2 . m i ); ii ) the computation of output signs s ij ; and iii ) the computation of the new a - posteriori probabilities λ j . the output messages from the cnp blocks are then multiplexed via multiplexer blocks 110 , 112 , and 114 to be written back at the proper addresses in the memory blocks a , s , and r . as illustrated , a check node module 130 comprises a plurality of check node processors cnp . the present invention is not limited to the embodiments described above . for instance , the foregoing detailed description has set forth various embodiments of the devices and / or processes via the use of block diagrams , schematics , and examples . insofar as such block diagrams , schematics , and examples contain one or more functions and / or operations , it will be understood by those skilled in the art that each function and / or operation within such block diagrams , flowcharts , or examples can be implemented , individually and / or collectively , by a wide range of hardware , software , firmware , or virtually any combination thereof . in one embodiment , the present subject matter may be implemented via asics . however , those skilled in the art will recognize that the embodiments disclosed herein , in whole or in part , can be equivalently implemented in standard integrated circuits , as one or more computer programs running on one or more computers ( e . g ., as one or more programs running on one or more computer systems ), as one or more programs running on one or more controllers ( e . g ., microcontrollers ) as one or more programs running on one or more processors ( e . g ., microprocessors ), as firmware , or as virtually any combination thereof , and that designing the circuitry and / or writing the code for the software and or firmware would be well within the skill of one of ordinary skill in the art in light of this disclosure . all of the above u . s . patents , u . s . patent application publications , u . s . patent applications , foreign patents , foreign patent applications and non - patent publications referred to in this specification and / or listed in the application data sheet , are incorporated herein by reference , in their entirety . from the foregoing it will be appreciated that , although specific embodiments of the invention have been described herein for purposes of illustration , various modifications may be made without deviating from the spirit and scope of the invention . accordingly , the invention is not limited except as by the appended claims .