As best I can tell from your description, the party who communicated with you is claiming trademark rights in "Lex-Code". There are many issues involved with such an assertion as you describe. Since your description is slightly confused, and they have threatened suit, you are best served consulting a trademark litigation lawyer, to whom you can show everything (i.e. your website, their communication, and anything else) without risk of waiving attorney-client privilege or work product immunity and they can do some review of relevant trademark and domain name registrations and then provide appropriate guidance to you.
If you are sure that it is a friend playig a practical joke, I suggest that you confront them before incurring any expenses with a lawyer. But act quickly, so you don't find yourself needlessly embroiled in a lawsuit if you are wrong.
This information is intended to be general and educational in nature. It is not intended to be specific legal advice or establish an attorney-client relationship. I require a signed retainer agreement from a potential client to establish an attorney-client relationship and before I will provide specific legal representation.
Assuming there is a company already doing business as Lex-Code then your registration and use of lexcode.co.uk is very likely in bad faith [and therefore "cybersquatting"] and that the company may, through a UDRP proceeding, force the registrar for that domain name to transfer it to the company. It is a different qustion whether your use of LexCode as the trade name of your company is also an infringement of the other company's trade name or trademark rights [permitting it to file a lawsuit against you]. No one can provide you with actionable advice without many more facts -- which you should only discuss with your own trademark attorney licensed to practice in your state. Good luck.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
It is a very fact specific question. They may or may not have the right to ask you to stop using the name. If, for example, they are in the US and you are in the UK unless they have procured rights in the UK they have no claim to the name there.
I would suggest you discuss it over with a lawyer in private or do your best to ascertain whether it is a legitimate cease & desist or not.
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Registration of a domain name does not give you any rights to use that name as a trademark. If someone else owns Lex-Code as a trademark and has been using it in commerce before you, that person owns the trademark and can demand that you cease using it as a domain name. The minor difference between LexCode and Lex-Code will not eliminate trademark infringement. The issue in most trademark disputes is whether there is a likelihood of consumer confusion. If your domain name choice could cause customers of Lex-Code to be confused as to the origin or source of goods and services, then you could be required to cease using the name. Unless this is a friend playing a trick, it sounds to me that you have a serious problem on your hands.
You made a fundamental mistake here---you chose a name for your business (and a domain name) without retaining legal counsel to conduct a trademark analysis. Failure to do so almost always leads to disastrous results---since you probably will required to cease using this name, choose a new one, and lose whatever investment you have made in the infringing name.
More generally, I can tell by your question that you do not have even a basic understanding of intellectual property law. In today's economy, this is a recipe for disaster. Your new business will fail unless you retain intellectual property counsel and lay the proper legal and IP law foundation for your new business. And meanwhile, you better spend a lot of time educating yourself on the basics of IP law. Look up patents, copyrights, and trademarks in Wikipedia and then read some articles and books on these issues. No responsible person can go into business in this century without acquiring a basic understanding of IP law.
In the U.S., the Anticybersquatting Consumer Protection Act ("ACPA") was enacted to protect consumers and holders of distinctive trademarks from "cybersquatting," which "involves the registration as domain names of well-known trademarks by non-trademark holders who then try to sell the names back to the trademark owners." Sporty's Farm L.L.C. v. Sportsman's Mkt., Inc., 202 F.3d 489, 493 (2d Cir. 2000). To successfully assert a claim under the ACPA, a plaintiff must demonstrate that (1) its marks were distinctive at the time the domain name was registered; (2) the infringing domain names complained of are identical to or confusingly similar to plaintiff's mark; and (3) the infringer has a bad faith intent to profit from that mark. See 15 U.S.C. § 1125(d)(1)(A). Webadviso v. Bank of Am. Corp., 448 Fed. Appx. 95, 97 (2d Cir. 2011).
If you registered the domain name in furtherance of your legitimate interests in the name LexCode (e.g., it is the name of your company or is a trademark of yours), then you did not have “bad faith” intent to profit from Lex-Code and Lex-Code has no cybersquatting claim in the U.S.