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I'm trying to understand why Gold codes and Kasami codes are used instead of pure m-sequences, in direct-sequence spread-spectrum (DSSS) communication systems, to prevent interference between multiple transmitters.

If I understand correctly, Gold codes are defined as the XOR between two m-sequences with different polynomials of the same degree (e.g. one LFSRs based on $p_1(x)$, and another LFSR based on $p_2(x)$ both of degree 20; the LFSR outputs are then XOR'd together), and within one system, multiple transmitters use the same pairs of polynomials but with a different time shift. (Kasami is also an XOR so I guess it's similar.)

How is this better than, say, having a system where all transmitters use m-sequences but each of them uses a different polynomial? (In both cases, the autocorrelation values are higher than the cross-correlation values)

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  • $\begingroup$ because the XOR and shift register stuffs are fast. $\endgroup$ – AlexTP Aug 15 '17 at 18:17
  • $\begingroup$ both are fast, but why are Gold/Kasami better? $\endgroup$ – Jason S Aug 15 '17 at 19:15
  • $\begingroup$ Activate and Accesses Processes into the Google Data Kasamai!Thank You! $\endgroup$ – Sherwin Esguerra Nov 24 '17 at 4:58
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There are very few m-sequences of any given length with good cross-correlation properties. Their autocorrelation properties are excellent, but the cross-correlation properties are variable. For example, there are 18 m-sequences of period 127 but to have good cross-correlation properties, one must choose a set of no more than $\require{cancel}\cancel{\text{3}}$ 6 m-sequences out of the 18 available. As a consequence, when several sequences with good cross-correlation properties are needed, one needs to use Gold sequence sets or Kasami sequence sets which have good cross-correlation properties and good (though not excellent) autocorrelation properties.

For more than you probably want to know, I refer you to the paper D.V. Sarwate and M.B. Pursley, "Cross-correlation properties of pseudorandom and related sequences," Proc. IEEE, vol.68, pp.593-619, May 1980. It includes a detailed discussion of the Gold sequences as well as the small sets and large sets of Kasami sequences.

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  • $\begingroup$ huh... what happens if you have 4 m-sequences? they have good pairwise cross-correlation properties, but .... ? $\endgroup$ – Jason S Aug 15 '17 at 19:04
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    $\begingroup$ I had typos in. my answer. At length 127, the cross-correlation between two m-sequences must have value at least 17. If you choose 6 m-sequences carefully (i.e. not a random selection), the $15 = \binom{6}{2}$ cross-correlations all have maximum value 17. If you add a 7th m-sequence into the set and it doesn't matter which the seventh one is, the new sequence will have cross-correlation at least 41 with one of the original 6 sequences. At longer lengths (up to $2^{15}-1$) there are only smaller subsets of m-sequences that will work, See Table 1 of my paper cited above. $\endgroup$ – Dilip Sarwate Aug 15 '17 at 19:52
  • $\begingroup$ Oh, I see, so the mutually acceptable m-sequences are hard to construct (need very particular subsets of m-sequences with a given length), whereas with Gold sequences they are easy and systematic to construct. $\endgroup$ – Jason S Aug 15 '17 at 20:01
  • $\begingroup$ are those pairs of m-sequences which have low cross-correlation the same thing as the "preferred pairs" found in your paper + others? $\endgroup$ – Jason S Aug 15 '17 at 20:17
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    $\begingroup$ Yes, and the sets of sequences in which each pair is a preferred pair is called a connected set. 6 is the size of the maximal connected set of length 127. $\endgroup$ – Dilip Sarwate Aug 15 '17 at 20:46

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